ࡱ> qX   ybjbj .xxqFF1111#1HDQ)l.WWWW &   C C C C C C C$EH>-C F   -CWWsD J"WWC C.r?0W0H3MQ1h.BD0HD.\HhH0?0H?0       -C-C   HD    H         F O: Lakireddy Bali Reddy College of Engineering, Mylavaram (Autonomous) Master of Computer Applications (I-Semester) MC105- Probability and Statistical Applications Lecture : 4 Periods/Week Internal Marks: 40 External Marks: 60 Credits: 4 External Examination: 3 Hrs. Faculty Name: N V Nagendram UNIT I Probability Theory: Sample spaces Events & Probability; Discrete Probability; Union, intersection and compliments of Events; Conditional Probability; Bayes Theorem . UNIT II Random Variables and Distribution; Random variables Discrete Probability Distributions, continuous probability distribution, Mathematical Expectation or Expectation Binomial, Poisson, Normal, Sampling distribution; Populations and samples, sums and differences. Central limit Elements. Theorem and related applications. UNIT III Estimation Point estimation, interval estimation, Bayesian estimation, Text of hypothesis, one tail, two tail test, test of Hypothesis concerning means. Test of Hypothesis concerning proportions, F-test, goodness of fit. UNIT IV Linear correlation coefficient Linear regression; Non-linear regression least square fit; Polynomial and curve fittings. UNIT V Queing theory Markov Chains Introduction to Queing systems- Elements of a Queuing model Exponential distribution Pure birth and death models. Generalized Poisson Queuing model specialized Poisson Queues. ________________________________________________________________________ Text Book: Probability and Statistics By T K V Iyengar S chand, 3rd Edition, 2011. References: 1. Higher engg. Mathematics by B V Ramana, 2009 Edition. 2. Fundamentals of Mathematical Statistics by S C Gupta & V K Kapoor Sultan Chand & Sons, New Delhi 2009. 3. Probability & Statistics by Schaum outline series, Lipschutz Seymour,TMH,New Delhi 3rd Edition 2009. 4. Probability & Statistics by Miller and freaud, Prentice Hall India, Delhi 7th Edition 2009. Planned Topics UNIT - II 1. Random Variables - Introduction 2. Discrete and Continuous Random Variables, Distribution Function 3. Mathematical Expectations, Examples 4. Problems 5. Binomial Distribution Mean, Variance, Mode 6. Problems 7. Poisson Distribution Mean, Variance, Mode 8. Tutorial 9. Normal Distribution Properties, Mean, Variance 10. Area under standard normal curve, Problems 11. Problems 12. Sampling distribution of mean 13. Sampling distribution of proportion 14. Sampling distribution of sum and differences 15. Central limit Theorem and Applications 16. Tutorial Chapter 2 Lecture 1 By N V Nagendram Probability Distributions ------------------------------------------------------------------------------------------ Introduction: In random experiments, we are interested in the numerical outcomes i.e., numbers associated with the outcomes of the experiment. For example, when 50 coins are tossed, we ask for the number of heads. Whenever we associate a real number with each outcome of trial, we are dealing with a function whose range is the set of real numbers we ask for such a function is called a random variable (r. v.) chance variable, stochastic variable or simply a variable. Definition: Quantities which vary with some probability are called random variables. Definition: By a random variable we mean a real number associated with the outcomes of a random experiment. Example: Suppose two coins are tossed simultaneously then the sample space is S= {HH, HT, TH, TT}. Let X denote the number of heads, then if X = 0 then the outcome is {TT} and P(X = 0) =  EMBED Equation.3 . If X takes the value 1, then outcome is {HT, TH} and P(X = 1) =  EMBED Equation.3 . Next if X takes the value 2 then the outcome is {HH} and P(X = 2) =  EMBED Equation.3 .The probability distribution of this random variable X is given by the following table: X = x012TotalP(X = x ) EMBED Equation.3  EMBED Equation.3  EMBED Equation.3 1 Example: out of 24 mangoes 6 are rotten, 2 mangoes are drawn. Obtain the probability distribution of the number of rotten mangoes that can be drawn: Let X denote the number of rotten mangoes drawn then X can take values 0, 1, 2.  EMBED Equation.3  EMBED Equation.3 ;  EMBED Equation.3  and  EMBED Equation.3  X = x012TotalP(X = x ) EMBED Equation.3  EMBED Equation.3  EMBED Equation.3 1 Types of Random Variables: There are two types of random variables: (i) Discrete random variables (ii) Continuous random variables Distribution function: Let X be a one-dimensional random variable. The function F defined for all x, by the equation F((x) = P(X ( x) is called the cumulative distribution function of X. Note: 1. We write c. d. f. For cumulative distribution function. Only d. f is written instead of c. d. f. Note 2: suffix X in F( is used to emphasize the fact that the distribution function is associated qith the particular valuiate X. when the particular underlying variate is clear from the context, we shall simply write F(x) insea of F((x). Note 3: tail events let x be any real number then the events |X < x | and |X> x|. |X ( x| are called tail events. For distinction, we may label them open, closed, upper and lower tails. Often, simple r.v.s are expanded as linear combination of tail events. Some Properties of a c. d. f.: P{a < X ( b} = F(b) F(a) Interval property 0 ( F(x) ( 1, ( x ( R Boundedness property F is non-decreasing i.e., if x ( y, then F(x) ( F(y) Monotone increasing property F(-() = 0, F(+() = 1 i.e., Lim F(xn) = 0 as n( - (; Lim F(xn) = 0 as n( ( Limits property F is continuous from the right each point Right continuous property i.e., F(a+) = F(a) F(a+) F(-a) = P(X = a) Jump discontinuity Conditions (3),(4) and (5) are necessary as well as sufficient for F to be c.d.f. on R. Problem 1: Give reasons why each of the graphs of F given below does not represent a distribution function. y=F(x) y=F(x) y=1 y=1   0 0 (a) (b) y=F(x) y=F(x) y=1 y=1    x = k 0 ( c ) 0 (d)  Solution: (a) F(x) < 0 ve for some x (b) F(x) > 1for some x ( c) F is non-decreasing i.e., some times F is decreasing also ( d )F is not right continuous at x = k infact it is left continuous. Definition: Discrete Random variables: Quantities which are capable of taking only integral values are called discrete random variables. Example: The number of children in a family of a colony. Example: The number of rooms in the houses of a township. Probability mass function: Probability distribution Definition: Let X be a discrete random variable taking value x, x = 0, 1, 2, 3, .... then P(X = x) is called the probability mass function of X and it satisfies the following ( i ) P(X = x) ( 0 ( ii )  EMBED Equation.3  Definition: Discrete distribution function: A r. v. X is said to be discrete, if there exist a countable number of points x1, x2, x3, . . . and number p(xi) ( 0,  EMBED Equation.3  such that  EMBED Equation.3 . Definition: Finite equiprobable space ( Uniform space) A finite equiprobable space is finite probability distribution where each sample point x1, x2, x3, . . .xn has the same probability for all i i.e., P(X = xi) = pi = a constant for all i and  EMBED Equation.3 . Chapter 2 Probability Distributions Tutorial 1 By N V Nagendram --------------------------------------------------------------------------------------------------------------- Problem 1: Show that the average of the deviations of a variate about its mean is zero and sum of the squared deviations is minimum when they are taken about the mean. [Ans. A=  EMBED Equation.3 ] Problem 2: A random variable X has the following probability distribution: x012345678P(x)k3k5k7k9k11k13k15k17kDetermine the value of k find P(X < 4), P(X ( 5), P(0 < x < 4) find the c.d.f. find the smallest value of x for which P(X ( x) < 0.5 [Ans. k= EMBED Equation.3  Problem 3:Given the discrete random variable X has mass function.  EMBED Equation.3  Describe and graph its cumulative distribution function, F(x). [Ans. 1/6,2/6 and 3/6] Problem 4: If  EMBED Equation.3  is a probability mass function, find F(x), the cumulative distribution function and sketch its graph. [Ans.  EMBED Equation.3 ] Problem 5: A random variables X has the following probability function. X = x01234567P(X= x)0k2k2k3kk22k27k2+1 Find the value of k P(X ( 5), P(X > 5), P(X < 6), P(X ( 6) P(0 < X < 6), P(0 < X < 5) [Ans. (i) k= - 1,  EMBED Equation.3 ; (ii) 0.81, 0.19 (iii) 0.81,0.8] Chapter 2 Probability Distributions Tutorial 2 By N V Nagendram --------------------------------------------------------------------------------------------------------------- Problem 1:A random variable X takes values 0, 1, 2 , 3, 4, . . . . with probability proportional to (x+1) EMBED Equation.3  [Ans. 0.9997] Problems 2: A random variable X has the following probability function: Value of x-2-10123P(x)0.1k0.22k0.3KFind the value of k, and calculate mean and variance. Construct the c.d.f. F(x) and draw its graph. [Ans. (i). 0.1,0.8 and 2.16 (ii). F(x) = 0.1,0.2,0.4,0.6,0.9,1.0] Problem 3: If a variable X assumes three values 0, 1, 2 with probabilities  EMBED Equation.3  respectively, find the c.d.f. of X and show that P(X ( 1) =  EMBED Equation.3 . Problem 4: A random variable X assumes the values -3, -2, -1, 0, 1, 2, 3 such that P(X > 0) = P(X = 0); P(X< -3) = P(X = - 2) = P(X = -1); P(X = 1) = P(X = 2) = P(X = 3) write down the distribution of X and show that P(X ( 3) =  EMBED Equation.3 . Definition: Expectation: The behaviour of r.v. either discrete or continuous is completely characterized by the distribution function F(x) or density f(x)[ P(xi) in discrete case( . instead of a function, a more compact description can be made by a single numbers such as mean (expectation), median and mode known as measures of central tendency of the r.v. X. Expectation or median or expected value of a r.v. X denoted by E[X( or (, is defined as  EMBED Equation.3  Definition: Variance: variance characterizes the variablility in the distributions, since two distributions with same mean can still have different dispersion of data about their means, Variance of r.v. X is  EMBED Equation.3  for X discrete  EMBED Equation.3  for X is continuous. Definition: Standard Deviation: standard deviation denoted by ( (S.D.) is the positive square root of variance.  EMBED Equation.3 (  EMBED Equation.3  (  EMBED Equation.3  (E(X2) - 2(.( ( (2.1 ( E(X2) - (2 since ( ((xf(x), (f(x)(1. Chapter 2 Lecture 2 By N V Nagendram Probability Density function (p. d. f) --------------------------------------------------------------------------------------------------------------- Let X be a continuous random variable taking value x, a ( x ( b then f9x) = P(X = x) is defined as the p. d. f. of X and satisfies the following (i)  EMBED Equation.3  ( ii)  EMBED Equation.3 . Note: 1. For a continuous variate, point probabilities are zero. 2. Area under the probability curve y = f(x) is unity; the fact  EMBED Equation.3 implies the graph f(x) is above x axis. 3. Area under the probability curve y = f(x) bounded by x = a, x = b is simply P(a ( x ( b ). 4. Relation between p. d. f. and c. d. f.: The density  EMBED Equation.3  and c. d. f. F are always connected by (a)  EMBED Equation.3  (b)  EMBED Equation.3 . Moments: If the range of the probability density function is from - ( to (, the rth moment about origin is defined as  EMBED Equation.3 . The r th moment about any arbitrary origin a is  EMBED Equation.3  The mean is given by (taking moment about x = 0)  EMBED Equation.3  The variance (2 is given by  EMBED Equation.3  Jointly Distributed Random Variables: Introduction: When the outcome of a random experiment can be characterized in more than one way, the probability density is a function of more than one variate. Example: When a card is drawn from an ordinary deck, it may be characterized according to its suit in some order viz., say clubs, diamonds, hearts and spades and Y be a variate that assumes the values 1, 2, 3, . . ., 13 which correspond to the denominations: Ace, 2, 3, . . ., 10, J, Q, K. Then (X, Y) is a 2 dimensional variate. The probability of drawing a particular card will be denoted by f(x, y) and if each card is equi-probable of being drawn, the density of (X, Y) is  EMBED Equation.3  EMBED Equation.3   EMBED Equation.3  Trails whose outcomes can be characterized by two (three) variates give rise to bivariate (tri-variate) distributions etc. Extensions to n-variate distributions are fairly straight forward. We study about the types of Distribution Functions as mentioned below: Joint distribution function - properties Joint discrete distribution function Individual or Marginal Probability functions Bivariate Probability distribution function Conditional proability functions Some properties of Joint density Individual or Marginal distribution functions Conditional distribution function Joint distribution Function and its properties: Let (X, Y) be a random vector or random variable on the probability space. The joint c. d. f. of X and Y is denoted by FX, Y and is defined by FX, Y(x, y) = P(Xd" x, Y d" y), x, y (( R.   S F(X, Y) c.d.f. Fig. A joint c. d. f. of two variates has the following properties: 1. Non-negativity and Boundedness: 0 d" FX, Y(x, y) d" 1, for every x, y (( R. 2. Monotonicity: the c. d. f.  F is monotonically non-decreasing function in each of the individual variables, i.e., ( i ) F(a, y2) e" F(a, y1), if y2 e" y1 ( ii ) F(x2, b) e" F(x1, b), if x2 e" x1. 3. Rectangle rule: Let a, b, c, d be any real numbers with a < b and c < d. Then, P(a < X d" b, c < Y d" d) = F(b, d) + F(a, c)  F(b, c)  F(a, d). 4. Individual limits: (i) Lim F(x, y) = F(- ", y) = 0 as n ((- ";(ii) EMBED Equation.3  EMBED Equation.3  5. Double limit:  EMBED Equation.3  i.e., F(",") = 1 formally. Note: we don t claim F(", y) = 1 or F(x, ") = 1. 6. Individual continuity: F is continuous from the right in each of its individual variables. i.e., (i)  EMBED Equation.3 , (ii)  EMBED Equation.3  7. If the density function f(x, y) is continuous at (x, y), then  EMBED Equation.3  Joint discrete Distribution Function: Definition: The joint c. d. f. of X and Y is said to be discrete if there exists a non-negative function P such that P vanishes everywhere except a finite or countably infinite number of points in the plane and at such points (x, y)so that P(x, y) = P( X = x, Y = y), for all x, y ( R. Definition: Let X and Y have a joint discrete distribution. A function P with does not vanish on the set {(xi, yi) such that I, j = 1, 2, 3, . . .} and satisfies the following properties: (i) P(xi, yi) e" 0 for all I, j = 1, 2, 3, . . . . . . and (ii)  EMBED Equation.3  is called joint probability (mass) function of X and Y or simply the joint probability function. Individual and Marginal Probability Functions: Let X and Y be two jointly distributed variables with joint discrete density P(x, y), the individual variates X and Y themselves are random variables. The individual distributions of X and Y are called marginal distributions of X and Y (i) The Marginal probability function for X is denoted by PX (x) or P(x) and is given by P(x) = P(X = x) =  EMBED Equation.3 =  EMBED Equation.3  (ii) The marginal probability function for Y is denoted by PY(y) and is given by P(y) = P(Y = y) =  EMBED Equation.3 =  EMBED Equation.3 . Note: It is convenient to display the probability function of a bivariate distribution in a rectangular array, in which the row totals and column totals provide the marginal probability functions of X and Y respectively. x yy1y2y3. . .yj . . .ym. . .P(X= xi )x1P11P12P13. . .P1j. . .P1m. . .P(x1)x2P21P22P23. . .P21. . .P2m. . .P(x2)x3. . .. . .. . .. . .. . .. . .. . .. . .. . ... . .. . .. . .. . .. . .. . .. . .. . .. . .xiPi1Pi2Pi3. . .Pij. . .Pim. . .P(xi)..........xnPn1Pn2Pn3. . .Pnj. . .Pnm. . .P(xn).. . .. . .. . .. . .. . .. . .. . .. . .. . .P(Y = yi )P(y1)P(y2)P(y3). . .P(yj). . .P(ym). . .1We have here, Pij = P(X = xi, Y = yj) ; P(xj) =  EMBED Equation.3 and P(yj) =  EMBED Equation.3 . Conditional Probability functions (cond. P. f.): Let X and Y have joint discrete distribution with associated probability function P. Let the possible values of X be {x1, x2, x3, . . .,xi, . . .} and those of Y be { y1, y2, y3, . . .,yj, . . .} respectively. The conditional probability function of X, given Y = yj denotd by P EMBED Equation.3  EMBED Equation.3  EMBED Equation.3  is defined by P EMBED Equation.3  EMBED Equation.3  EMBED Equation.3  = P(xi, yj)/PY(yj) for i = 1, 2, 3, . . . = 0 if PY(yj) = 0 The conditional probability function of Y, given X = xi denoted by P EMBED Equation.3  EMBED Equation.3  EMBED Equation.3  is defined by P EMBED Equation.3  EMBED Equation.3  = P(xi, yj)/PX(xi) for j = 1, 2, 3, . . . = 0 if PX(xi) = 0 Therefore, P(xi, yi) = P(X = xi, Y = yj) ; P(Y = yj) = PY(yj) and P(X = xi) = PX (xi) Independent and dependent Random Variables: Definition: Two random variables x and Y are called independent if for every pair of real number  x and  y , the two events {X d" x} and {Y d" y} are independent. That is we can express as P{ X d" x, Y d" y} = P{ X d" x} P{Y d" y} ------------------------------------------ (1) Condition (1) in terms of distribution functions is F(x, y) = FX (x) FY (y)-------------------(2) and also  EMBED Equation.3  [if densities exists]-----------------------------------------------(3) Conversely, if (2) or (3) is true, then (1) follows. Definition: dependent Variates: Variates which are not independent are called dependent variates or dependent random variables. Definition: continuous random Variates: A 2-dimensional random vector (X, Y) is called a continuous random vector if there exists a function f(x, y) e" 0 such that for - " < x, y < ", the c. d. f. of (X, Y) given by  EMBED Equation.3  is continuous. The function f(x, y) is called the joint p. d. f of (X, Y). Some properties of joint density: Let f(x, y) e" 0 be the joint p. d. f of continuous random vector (X, Y) and F(x, y) be the c. d. f. of (X, Y) then it holds the following properties: (i)  EMBED Equation.3  (ii) P{ a < X d" b, c < Y d" d} =  EMBED Equation.3  (iii)  EMBED Equation.3  Individual or Marginal Distributions: Let (X, Y) be a continuous random vector with joint c. d. f. F and joint p. d. f. f. Then F(x, y) = P(X ( x, Y ( y) =  EMBED Equation.3 . Definition: Let (X, Y) be a 2-dimesnional continuous random vector with joint p. d. f. f(x, y). Then the individual or marginal distribution of X and Y are defined by the p. d. f. s  EMBED Equation.3  and  EMBED Equation.3 . On observation, we have EMBED Equation.3 . Conditional Distribution Function: The conditional c. d. f. of a variate X, given Y = y, written  EMBED Equation.3  ------------------------------------------(1) Provided that the limit in (1) exists. The conditional p. d. f. of X given Y = y, written  EMBED Equation.3  is a non-negative function satisfying EMBED Equation.3 . Note: The conditional p. d. f. f(x/y) is given by fX/Y(x/y) = f(x, y)/fY(y) where fY is the marginal p. d. f. of Y, fY(y) > 0, and is continuous. Chapter 2 Lecture 3 Probability Distributions Lecture by N. V. Nagendram ---------------------------------------------------------------------------------------------------------------- Mathematical Expectation: Let X be a discrete random variable taking value x, x = 0, 1, 2, 3, . . . . then the mathematical expectation of X is denoted by E(x) [read it as expected value of X] and is defined as  EMBED Equation.3  EMBED Equation.3  Similarly if k is any positive integer then,  EMBED Equation.3 . Similarly if X is continuous random variable taking value x, - " ( x ( " with f(x) as the probability density function then  EMBED Equation.3  and  EMBED Equation.3 . Definition: E(X) is also called mean or arithmetic mean of X denoted by . Definition: If X is a random variable then the variance of X ise denoted by V(X) and is defined as V(X) = E[(X E(X))2]. This can be simplified as V(X) = E(X2) [E(X)]2. Notation: The variance is denoted by (2 = V(X). Standard deviation: The positive square root of variance is defined as standard deviation and is denoted by (. Therefore,  EMBED Equation.3 . Theorem of Expectation of a sum and product: Expectation of the sum of the random variables: Theorem: The mathematical expectation of a sum of a number of random variables is equal to the sum of their expectations. Proof: follows Expectation of the product of the random variables: Theorem: The mathematical expectation of the product of a number of independent random variates is equal to product of their expectations. Theorem: The mathematical expectation of a sum of a number of random variables is equal to the sum of their expectations. Proof: Let us consider the random variables x and y. Let x assume the values xi for all I = 1, 2, 3, . . .,m and y the values yj for all j = 1, 2, 3, . . . ,n with respective probabilities Pi and Pj. The sum x + y is a random variable which can take m n values, xi + yj for i = 1,2,3,.,m for j = 1,2,3,..,n with probabilities Pij. Hence its Expectation is  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3 = E(x) + E(y) (( E(x + y) = E(x) + E(y) Since  EMBED Equation.3  and  EMBED Equation.3 . By generalization of the above theorem, we have E(x1 + x2 +x3 + . . .+xn) = E(x1) + E(x2) + E(x3) . . . . .+ E(xn). This completes the proof of the theorem.( Expectation of the product of the random variables: Theorem: The mathematical expectation of the product of a number of independent random variates is equal to product of their expectations. Proof: Let us use the notation as  EMBED Equation.3  Since the variates are independent, by the law of compound probabilities we have  EMBED Equation.3   EMBED Equation.3  The theorem can be generalized for a number of independent random variates such that E(x1. x2. x3 . . . . . xn) = E(x1) . E(x2) . E(x3) . . . . .. E(xn). This completes proof of the theorem.( Note: E(x, y) = E(x) E(y) does not guarantee the independent of x and y. One can easily verify that the following expectations: (a) E(a) = a (b) E(aX) = a E(X) (c) E(aX bY) = aE(X) bE(Y) (d) E(aX + b) = a E(X) + b (e ) V(a) = 0 (f) V(aX b) = a2 V(X) (g) V(x) = E(x2) [E(x)]2 (h) V(aX + bY) = a2 (2X + b2Y + 2ab (XY. Chapter 2 Probability Distributions Tutorial 3 By N V Nagendram --------------------------------------------------------------------------------------------------------------- Problem 1: Two coins are tossed simultaneously. Let X denote the number of heads, Find E(X) and V(X)? Solution: X = x012TotalP(X = x) EMBED Equation.3  EMBED Equation.3  EMBED Equation.3 1 Mean: ( = E(X) = 0.  EMBED Equation.3  + 1.  EMBED Equation.3  + 2.  EMBED Equation.3  = 1 Variance: (2 = V(X) = E(X2) [E(X)]2 = 02.  EMBED Equation.3  +12.  EMBED Equation.3  + 22.  EMBED Equation.3  - (1)2 =  EMBED Equation.3  + 1 1 =  EMBED Equation.3  Hence the solution. Problem 2: If it rains, a dealer in rain coats earns Rs. 500/- per day and if it is fair, he loses Rs.50/- per day. If the probability of a rainy day is 0.4. Find his average daily income? Solution: X = x500-50TotalP(X = x)0.40.61 Average = E(X) = 500 (0.4) + (-50) (0.6) = 200 30 = Rs. 170/- Hence the solution. Chapter 2 Lecture 4 Probability Distributions Lecture by N. V. Nagendram ---------------------------------------------------------------------------------------------------------------- Binomial Distribution: This distribution was discovered by James Bernoulli. This is a discrete distribution. It occurs in cases of repeated trials such as students writing an examination, births in a hospital etc. Here all the trials are assumed to be independent and each trial has only two outcomes namely success and failure. Let an experiment consist of n independent trials. Let it succeed x times. Let p be the probability of success and q be the probability of failure in each trial. ( p + q = 1 The probability of getting x successes = p.p.p............p(x times) = px The probability of getting (n x) failures = q.q.q...........q[(n x) times] = q(n x) ( From multiplication theorem, the probability of getting x successes and (n x) failures is px q(n x). This is the probability of getting x successes in one combination. There are such nCx mutually exclusive combinations each with probability px q(n x). ( From addition theorem the probability of getting x success in nCx px q(n x). Notation: b(x; n, p) denotes a binomial distribution with x successes, n trials and with p as the probability of success. ( b(x; n, p) = nCx px q(n x), x = 0, 1, 2, 3, . . . ., n. Parameters of Binomial distribution: In b(x; n, p) there are 3 constants viz., n, p and q. Since q = 1- p, hence there are only 2 independent constants namely n and p. These are called the parameters of binomial distribution. Note: since b(x; n, p) is same as the (x + 1)th term in the binomial expansion of (q + p)n, hence this distribution is called the Binomial Distribution. Mean of the Binomial distribution:  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  Put y = x 1, ( x = 1 + y When x = 1 implies y = 0 x = 1 implies y = x 1  EMBED Equation.3   EMBED Equation.3  So, ( = np is Mean of Binomial distribution. Variance of the Binomial Distribution:  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  Put y = x 2 ( x = 2 + y When x = 2 implies y = 0 When x = n imples y = n - 2  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  ( The variance of binomial distribution is npq. The standard deviation is ( = + EMBED Equation.3 . Moments of Binomial distribution: Mean  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  by definition  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  Consider  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  (  EMBED Equation.3  (2 (variance) =  EMBED Equation.3  = n(n 1) p2 + np (np)2  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  [since, 1 p = q] ( np > npq [since q is a fraction] Mean > Variance Similarly (3 = npq(1 2p) Hence (1 =  EMBED Equation.3  When p =  EMBED Equation.3 = q Therefore (1 = 0 Case (i) when p =  EMBED Equation.3 , (1 = 0 Case (ii) when n ( ( then (1 = 0 Standard deviation =  EMBED Equation.3  Skewness =  EMBED Equation.3  Moment Generating Function of Binomial Distribution: Let X be a variable following binomial distribution, then  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  M.G.F about Mean of binomial Distribution:  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  Since we have, a3+b3 = (a+b)(a2 ab + b2) ; p3+q3 = (p + q)(p2 pq + q2) = (1)(p+q)2 3(pq) = (1 3pq) Now (2 = coefficient of  EMBED Equation.3 ; (3 = coefficient of  EMBED Equation.3 ; (4 = coefficient of  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  Additive Property of Binomial Distribution: Let X ( b(n1, p1) and Y ( b(n2, p2) be independent random variables. Then  EMBED Equation.3 ;  EMBED Equation.3  what is distribution of X + Y We have  EMBED Equation.3 [ since X and Y are independent] = (q1 + p1 et)n1 (q2 + p2 et)n2 Since (2) cannot be expressed in the form (q + pet)n , from uniqueness theorem of m.g.f it follows that X + Y is not a binomial variate. Hence, in general the sum of two independent binomial variates is not a binomial variate. In other words, binomial distribution does not possess the additive or reproductive property. However, if we take p1 = p2 = p say then from (2), we get  EMBED Equation.3  which is the m.g.f of a binomial variate with parameters (n1+n2, p). Hence, by uniqueness theorem of m.g.f s X + Y ( b(n1+n2,p). Thus the binomial distribution possesses the additive or reproductive property if p1 = p2. Generalization: If Xi for all i = 1, 2, 3, . . . ,k then their sum  EMBED Equation.3  Recurrence Relation for the probabilities of Binomial Distribution: (Fitting of Binomial Distribution) We have  EMBED Equation.3  EMBED Equation.3  and  EMBED Equation.3  ( EMBED Equation.3   EMBED Equation.3  (  EMBED Equation.3  which is the required recurrence formula. *** *** *** *** *** *** *** *** *** *** *** *** Chapter 2 Probability Distributions Tutorial 4 By N V Nagendram --------------------------------------------------------------------------------------------------------------- Problem 1: It has been claimed that in 60% of all solar heat installations the utility bills is reduced by at least one third. Accordingly what are the probabilities that the utility bill will be reduced by at least one third in (i) four or five installations (ii) at least four of five installations? Problem 2: Two coins are tossed simultaneously. Find the probability of getting at least seven heads? Problem 3: If 3 of 20 tyres are defective and 4 of them are randomly chosen for inspection. What is the probability that only one of the defective tyres will be included? Chapter 2 Probability Distributions Tutorial 4 By N V Nagendram --------------------------------------------------------------------------------------------------------------- Problem 1: It has been claimed that in 60% of all solar heat installations the utility bills is reduced by at least one third. Accordingly what ae the probabilities that the utility bill will be reduced by at least one third in (i) four or five installations (ii) at least four of five installations? Solution: n = 5, p = 0.6, q = 1 p = 0.4 b(4; 5, 0.6) = 5C4 (0.6)4 (0.4)1 = 5(0.6)4(0.4) = 0.2592 at least 4 means 4 or 5 b(5; 5, 0.6) = 5C5 (0.6)5 (0.4)0 = 0.0778 ( Probability in at least four installations = b(4; 5, 0.6) + b(5; 5, 0.6) = 0.2592 + 0.0778=0.337 Hence the solution. Problem 2: Two coins are tossed simultaneously. Find the probability of getting at least seven heads? Solution: n = 10, p = P(H) =  EMBED Equation.3 ; q = 1 p =  EMBED Equation.3  P(X ( 7) = P(X = 7) + P(X = 8) + P(X = 9) P(X = 10) = 10C7(1C2)7 (1C2)3 + 10C8 (1C2)8 (1C2)2 + 10C9 (1C2)9 (1C2)1 + 10C10 (1C2)10 (1C2)0 =  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3 = 0.172 Hence the solution. Problem 3: If 3 of 20 tyres are defective and 4 of them are randomly chosen for inspection. What is the probability that only one of the defective tyres will be included? Solution: n = 4, p =  EMBED Equation.3 , q = 1- p =  EMBED Equation.3  P(x = 1) = 4C1 (p)1 (q)(4 - 1) = 4  EMBED Equation.3  Hence the solution. Chapter 2 Probability Distributions Tutorial 5 Binomial distribution By N V Nagendram --------------------------------------------------------------------------------------------------------------- Problem 1: Determine the binomial distribution for which the mean is four and variance three. Also find its mode? [Ans.4.25or4] Problem 2: If A and B play games of chess of which 6 are won by A, 4 are won by B and 2 end in draw. Find the probability that (i) A and B win alternatively (ii) B wins at least one game (iii) Two games end in draw? [Ans.5/36,19/27,5/72] Problem 3: If the probability that a person will not like a new tooth paste is 0.20. what is the probability that 5 out of 10 randomly selected persons will dislike it? [Ans. 0.0264] Problem 4: A shipment of 20 tape recorders contains 5 defectives find the standard deviation of the probability distribution of the number of defectives in a sample of 10 randomly chosen for inspection? [Ans,(= EMBED Equation.3  Problem 5: If A and B play game in which their chances of winning are in the ratio 3 : 2 Find As chance of winning at least three games out of the five games played? [Ans. 0.68] Problem 6: A department has 10 machines which may need adjustment from time to time during the day. Three of these machines are old, each having a probability of  EMBED Equation.3 of needing adjustment during the day and 7 are new, having corresponding probabilities of  EMBED Equation.3 . Assuming that no machine needs adjustments twice on the same day, determine the probabilities that on a particular day. (i) just 2 old and no new machines need adjustment. (ii) if just 2 machines need adjustment, they are of the same type. [Ans. 0.016;0.028] Problem 7: An irregular six faced die is thrown and the probability exception that in 10 throws it will give five even numbers is twice, the probability expectation that it will give four even numbers. How many times in 10000 sets of 10 throws each, would you expect it to give no even number? [Ans. 1 approxly] Problem 8: The mean of binomial distribution is 3 and variance is 4? [Ans.  EMBED Equation.3 ] Problem 9: The mean and variance of binomial distribution are 4 and  EMBED Equation.3  respectively. Find P(X ( 1)? [Ans.0.9983] ********* Chapter 2 Probability Distributions Tutorial 6 Binomial distribution By N V Nagendram --------------------------------------------------------------------------------------------------------------- Problem 01: Find a binomial distribution for the following data and compare the theoretical frequencies with the actual ones: x:012345f:2142034228 [Ans.100(0.432 + 0.568) Problem 02: The probability that a bomb dropped from a plane will strike the target is  EMBED Equation.3 . If six bombs are dropped, find the probability that (i) exactly two will strike the target, (ii) at least two will strike the target. [Ans. (i) 0.246 (ii)0.345] Problem 03: If the probability that a new-born child is a male is 0.6, find the probability that in a family of 5 children there are exactly 3 boys? [Ans. 0.3456] Problem 04: Find the probability of guessing correctly at least 6 of the 10 answers on a true-false examination? [Ans.  EMBED Equation.3 ] Problem 05: Out of 800 families with 5 children each, how many would you expect to have (i) 3 boys (ii) 5 girls and (iii) either 2 or 3 boys? Assuming that equal probabilities for girls and boys. [Ans.(i)250 (ii) 25 (iii) 500] Problem 06: If the probability of a defective bolt is 0.1, find (i) the mean and (ii) the standard deviation for the distribution of defective bolts in a total of 400? [Ans. (i) 40 (ii) 6] Problem 07: Find the probability that in five tosses of a fair die a 3 appears (i) at no times (ii) four times? [Ans. (i)  EMBED Equation.3 (ii)  EMBED Equation.3 ] Problem 08: Find the probability that in a family of 4 children there will be (i) at least 1 boy and (ii) at least 1 boy and 1 girl? [Ans. (i)  EMBED Equation.3 (ii)  EMBED Equation.3 ] Problem 09: Find the probability of getting at least 4 heads in 6 tosses of a fair coin? [Ans.  EMBED Equation.3 ] Chapter 2 Probability Distributions Tutorial 7 Binomial distribution By N V Nagendram --------------------------------------------------------------------------------------------------------------- Problem 1: The following data due to Weldon shows the results of throwing 12 dice 4096 times, a throw of 4, 5 or 6 being called success (x). X0123456789101112V-7601984307319488475362577111-Fit a Binomial distribution and calculate the expected frequency? [Ans.  EMBED Equation.3 ] Problem 2: Fit a Binomial distribution to the following data and test for goodness of fit X01234F286246104 [Ans.  EMBED Equation.3 ] Problem 3: In 256 sets of 12 tosses of a coin, in how many cases one can expect eitght heads and 4 tails? [Ans.P(X=8)= EMBED Equation.3 ] Problem 4: The mean and variance of a binomial variate X with parameters n and p are 16 and 8. Find (i) p(X = 0) (ii) p(X = 1) and (iii) p(X ( 2). [Ans. (i) p(X = 0) = 32C0  EMBED Equation.3 ; (ii) P(X = 1) = 32C1  EMBED Equation.3 ; And (iii) P(X ( 2) = 1 {32C0  EMBED Equation.3 +32C1  EMBED Equation.3 ] Problem 5: Seven coins are tossed and the number of heads are noted. The experiment is repeated 128 times and the following distribution is obtained: No of heads01234567TotalFrequencies761935302371128 Fit a binomial distribution(B.D.) assuming (i) the coin is un biased (ii) the nature of the coin is not known? Chapter 2 Lecture 5 Chebyshevs theorem Probability Distributions by N. V. Nagendram Chebyshevs theorem: Let X be a random variable with mean ( and standard deviation ( then P(| x - ( | ( k() (  EMBED Equation.3 . Proof: Let f(x) be the probability mass function of a random variable having mean ( and variance (2. Now  EMBED Equation.3  .(1) Let R1 be the region in which x ( ( - k(, R2 the region in which ( - k( < x < ( + k( and R3 be the region in which x ( ( + k(.  x ( ( - k( ( - k( < x < ( + k( x ( ( + k(   R1 region R2 Region R3 Region Values of x (  EMBED Equation.3  (2) Since  EMBED Equation.3 ( 0 i.e., non-negative, hence  EMBED Equation.3  ( 0 also non-negative. So equation (2) implies  EMBED Equation.3  .(3) In R1 x ( ( - k( ( x - ( ( k( .(4) In R3 x ( ( + k( ( x - ( ( k( .(5) Equations (4) and (5) ( | x - ( | ( k( Hence both R1 and R3,  EMBED Equation.3 ( k2(2. ( From (3)  EMBED Equation.3  .(6) i.e.,  EMBED Equation.3 (  EMBED Equation.3 ...(7) Now  EMBED Equation.3  represents the probability assigned to the region R1 ( R3. (  EMBED Equation.3  = P[| x - ( | ( k(] .(8) ( From equations (7) and (8)  EMBED Equation.3  (  EMBED Equation.3  This completes the proof of the theorem Note:  EMBED Equation.3  ( EMBED Equation.3 . Chapter 2 Probability Distributions Tutorial 8 Chebyshevs theorem By N V Nagendram --------------------------------------------------------------------------------------------------------------- Problem 1: X is random variable such that E(X) = 3 and E(X2) = 13. Determine a lower bound for P( - 2 < x < 8), using Chebyshev s inequality? [Ans. ( = 2; lower bound =  EMBED Equation.3 ] Problem 2: 500 articles were selected at random out of a batch containing 10000 articles and 30 were found to be defective. How many defective articles would you reasonably expect to have in the whole batch? [Ans. E(X)=Np=10000X EMBED Equation.3 ] Problem 3: A symmetric die is thrown 600 times. Find the lower bound for the probability of getting 80 to 120 sixes? [Ans. P(80 ( x ( 120 =  EMBED Equation.3 ] Problem 4: Given that the discrete random variable X has density function f(x) given by f(-1)=  EMBED Equation.3 , f(0) =  EMBED Equation.3 , f(1) =  EMBED Equation.3  use Chebyshev s inequality to find the upper bound when k = 2? [Ans.  EMBED Equation.3 ] Problem 5: For geometric distribution P(x) = 2-x; x = 1, 2, . . . .Prove that Chebyshev s inequality gives P[(| x - 2 |) ( 2] >  EMBED Equation.3 while the actual probability is  EMBED Equation.3 . Problem 6: Two unbiased dice are thrown. If X is the sum of the numbers showing up. Prove that P[(| x - 7 |) ( 3] (  EMBED Equation.3  Also compare this with actual probability? Problem 7: Suppose that X assumes the values 1 and 1, each with probability 0.5. Find and compare the lower bound on P[ -1 < X < 1] given by Chebyshev s inequality and the actual probability that 1 < X < 1? Problem 8: Find a lower bound on P[ - 3 < X < 3] where ( = E(X) = 0 and variance =(2 = 1. [Ans. L.b =  EMBED Equation.3 ] Problem 9: Use Chebyshevs inequality to find a lower bound (l. b.) on P[ -4 < X < 20 ] where the random variable X has a mean ( = 8 and variance (2 = 9. [Ans.  EMBED Equation.3 ] Problem 10: If X is the number appearing on a die when it is thrown, show that the Chebyshev s theorem gives P[| x - (| > 2.5] < 0.47 while the actual probability is zero. Problem 11: The number of customers who visit a car dealer show room on a certain day is a random variable with mean 18 and standard deviation 2.5. With what probability can it be asserted that there will be between 8 and 28 customers? [Ans.  EMBED Equation.3 ] Chapter 2 Probability Distributions Tutorial 9 Chebyshevs theorem By N V Nagendram --------------------------------------------------------------------------------------------------------------- Problem 1: A random variable X has density function given by  EMBED Equation.3  (a) Find P[| x - ( | > 1 ]; (b) Use Chebyshevs inequality to obtain an upper bound on P[| x - (| > 1] and compare with the result in (a). [Ans. (a) e-3 = 0.04979 (b) 0.25] Problem 2: Prove Chebyshev s inequality for a discrete variable X? Problem 3: Let X1, X2, X3, . . . ,Xn be n independent random variables each having density function  EMBED Equation.3 . If Sn = X1+X2+X3+ . . . ,+Xn then show that P EMBED Equation.3 . Problem 4: A random variable X has mean 3 and variance 2. Use Chebyshev s inequality to obtain an upper bound for (a) P[| X 3| ( 2] (b) P[| X (| ( 1] [Ans. 1, EMBED Equation.3 ] Problem 5: A random variable X has the density function  EMBED Equation.3  then (a) find P[| X (| ( 2] (b) use Chebyshevs inequality to obtain an upper bound on P[| X (| ( 2] and compare with the result in (a). [Ans.(a)e-2, (b) 0.5] Chapter 2 Lecture 6 Poissons theorem Probability Distributions by N. V. Nagendram Definition: A random variable X is said to follow Poisson distribution if its probability mass function is given by f(x, () =  EMBED Equation.3 , for x = 0, 2, 3, . . . = 0 otherwise. Poisson Approximation to Binomial Distribution Theorem: Statement: As n ( ( and p ( 0 so that np = ( where ( is a finite non-zero constant then b(x, n, p)(  EMBED Equation.3 . Proof: Let us consider b(x, n, p) so that  EMBED Equation.3   EMBED Equation.3  and given np = ( (  EMBED Equation.3  also q = 1- p = 1 -  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  EMBED Equation.3 ---------- (1) Now as n ( (,  EMBED Equation.3   EMBED Equation.3  and  EMBED Equation.3  ( from equation (1) b(x, n, p) (  EMBED Equation.3 . This completes the proof of the Poissons Approximation to Binomial distribution theorem. Note: 1.  EMBED Equation.3  2. Show that  EMBED Equation.3  For that consider  EMBED Equation.3  3. ( > 0 is called the parameter of the Poisson Distribution. 4. P(X = 0) =  EMBED Equation.3  Applications of Poisson distribution: Poisson distribution is applicable when n is very large and p is very small. Hence some of the applications of Poisson distribution are as follows: 1. Number of faulty blades produced by a reputed firm 2. Number of deaths from a disease such as heart attack or cancer. 3. Number of telephone calls received at a particular telephone exchange. 4. Number of cars passing a crossing per minute. 5. Number of printing mistake in a page of a book. Mean and Variance of Poisson distribution: Mean ( = E(X)  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  Therefore Mean = ( = (  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  ( E(X2) = (2 e-(e( + ( E(X2) = (2 + ( Variance = V(X) = (2 = E(X2) [E(X)]2 = (2 + ( - (2 = (. ( variance = ( Standard Deviation = S.D. =  EMBED Equation.3  Note : In a Poisson distribution mean always equal to the variance. Chapter 2 Lecture 7 Poissons m. g. f. Probability Distributions by N. V. Nagendram --------------------------------------------------------------------------------------------------------------- Moment Generating Function of Poisson Distribution: MX(t) = E[etx] =  EMBED Equation.3  Additive Property of Poisson Variates: Theorem: If x and Y are two independent random Poisson variates with parameters ( and ( then X + Y is also a Poisson variate with parameter ( + (. Proof: Since X is a Poisson variate with parameter ( ( MX(t) =  EMBED Equation.3  Similarly, since Y is Poisson variate with parameter ( ( MY(t) =  EMBED Equation.3  From the additive property of the moment generating function MX+Y (t) = MX(t). MY (t) =  EMBED Equation.3 . EMBED Equation.3  =  EMBED Equation.3  Which is the moment generating function of a Poisson variate with parameter ( + ( . ( X + Y is also a Poisson variate with parameter ( + ( . POISSON PROCESS:  (t  T ( = np ( p =  EMBED Equation.3  ( = np =  EMBED Equation.3  P(X=x) = [e-(t ((T)x ]/x! Suppose we have to find the probability of x successes during a time interval T. Divide the time interval T into n equal parts of width (t. Therefore T = n. (t . Here we make the following assumptions: (a) The probability of success during an interval (t is given by (.(t. (b) The probability more than one success in a small interval (t is negligible. (c) The probability of success in interval (t, t+(t) is independent of the actual time t and also of all previous successes. Here the assumptions of binomial distribution are satisfied and the probability b(x, n, p) where  EMBED Equation.3 . ( As n (( binomial distribution approaches to Poisson distribution and here parameter ( is ( = np ( p =  EMBED Equation.3  ( = np =  EMBED Equation.3  ( P(X=x) = [e-(t ((T)x ] / x! Chapter 2 Lecture 8 Normal distribution Probability Distributions by N. V. Nagendram --------------------------------------------------------------------------------------------------------------- Normal Distribution (N.Dn): Normal distribution is also a continuous distribution. A random variable X is said to follow normal distribution (N. Dn) with mean ( and variance (2 if its probability density function is given by  EMBED Equation.3 , -( < x < ( ; -( < ( < ( ; ( > 0 = 0 , otherwise. The corresponding distribution function is  EMBED Equation.3  Let Z =  EMBED Equation.3  then the mean of Z is 0 and the variance is 1. The corresponding probability density function is  EMBED Equation.3  Z is called standard normal variate. Notation : 1. X ( N((, (2) denotes that X is a normal variate with mean ( and variance (2. 2. Z ( N(0, 1). Features of Normal Distribution curve: The graph of f(x) is a bell shaped curve extending from - ( to ( with its peak at (.   - ( ( ( Graph of ((Z):   - ( ( ( Note 1. The mode of normal distribution is (. 2. The median of normal distribution is also (. Hence for a normal distribution the mean, median and mode coincide.       The area under the normal curve between the ordinates x = a and x = b gives the probability that the random variable X lies between a and b. ( P(a < X < b) =  EMBED Equation.3  Put Z =  EMBED Equation.3  So dz =  EMBED Equation.3  ( dx = (. dz When x =a , z =  EMBED Equation.3 =c (say) When x = b, z =  EMBED Equation.3 =d (say) ( P(a < X < b) =  EMBED Equation.3 . So, 1.  EMBED Equation.3  2.  EMBED Equation.3  3.  EMBED Equation.3  4.  EMBED Equation.3  The  EMBED Equation.3 are available in the table 1. Chapter 2 Probability Distributions Tutorial 10 Normal Distribution to B. D.n By N V Nagendram --------------------------------------------------------------------------------------------------------------- Problem 1# Prove that normal distribution is a limiting form of Binomial distribution? Problem 2# If 20% of the memory chips made in a certain plant are defective what are the probabilities that in a lot of 100 randomly chosen for inspection ( i) at most 15 will be defective ( ii) exactly 15 will be defective. [Ans. i) 0.1292 ii) 0.0454] Problem 3# The mean weight of 500 male students at a certain college is 75 kg and the standard deviation is 7 kg. Assuming that the weights are normally distributed. Find how many students weigh (i) between 60 and 78 kg (ii ) more than 92 kg. [Ans. 0.4838+0.1664=0.6502 ii) 0.5000-0.4925 = 0.0075] Problem 4# Find the probability of getting 3 and 6 heads inclusive in 10 tosses of a fair coin by using (i) Binomial distribution (ii) the normal approximation to the binomial distribution. [Ans. 0.773 ; 0.6337] Problem 5# If the masses of 300 students are normally distributed with mean 68.0 kg and standard deviation 3.0 kg, how many students have masses: (i) 72 kgs (ii) ( 64 kgs (iii) 65 ( X ( 71 kg inclusive [Ans. i)0.0918 28 students ii) 0.0918 28 students iii) 0.6826 205 students] Chapter 2 Probability Distributions Tutorial 11 Poissons By N V Nagendram --------------------------------------------------------------------------------------------------------------- Problem 1# Define Poisson process with example and show that mean = variance for a Poisson distribution? Solution: Definition: Poisson process: The Poisson process is the method of obtaining Poisson distribution independently without considering it as a limiting case of binomial distribution. It will be a Poisson distribution with parameter (t. Example: 1. No. of telephones were Poisson process at a telephone exchange 2. No. of deaths due to heart attack or cancer. To show that mean = variance in a Poisson distribution. For that Consider ( = E(X) =  EMBED Equation.3 =  EMBED Equation.3  ( ( = ( Consider E(X2) =  EMBED Equation.3   EMBED Equation.3  =  EMBED Equation.3  = (2e-(.e( + ( ( E(X2) = (2 + ( and (2 = V(X) = E(X2) [E(X)]2 = (2 + ( - (2 ( (2 = (. ( ( = (2 i.e., mean = variance Hence the solution. Problem 2# If the probability that an individual suffers a bad reaction due to a certain injection is 0.001, determine the probability that out of 2000 individuals (i) exactly 3 (ii) more than 2 individuals will suffer a bad reaction? Solution: Given p = 0.001 ; n = 2000 ; ( = np = 2 (i) to find P(Exactly 3) = P(X=3) =  EMBED Equation.3  since e=2.086, 2 2) = 1 P(X( 2) = 1 [P(X=0) +P(x=1) + P(x=2)] = 1 [ EMBED Equation.3 + EMBED Equation.3 + EMBED Equation.3 ] = 1 e-( [1+(+ EMBED Equation.3 ] = 1 5e-2 = 0.323. Hence the solution. Problem 3#A manufacturer of cotter pins knows that 5% of his product is defective. If he sells cotter pins in boxes of 100 and guarantees that not more than 10 pins will be defective, what is the approximate probability that a box will fail to meet the guaranteed quality? Solution: We are given n = 100, p = probability of defective pin = 5% = 0.05 And ( = mean number of defective pins in a box of 100 = np = 100 X 0.05 = 5 Since p is small, we may use Poisson distribution probability of x defective pins in a box of 100 is P(X=x)  EMBED Equation.3  Probability that a box will fail to meet the guaranteed quality is P(X> 10) = 1- P(X( 10) = 1 -  EMBED Equation.3  = 1 e-5  EMBED Equation.3  Hence the solution. Problem 4# 10% of the bolts produced by a certain machine turn out to be defective. Find the probability that in a sample of 10 tools selected at random exactly two will be defective using (i) binomial distribution (ii) Poisson distribution and comment upon the result? Solution: Given p =  EMBED Equation.3 , n = 10, ( = np = 1 (i) Using binomial distribution Let q = 1 p = 1 0.1 = 0.9 P(X=2) = 10C2 p2 q(n -2) =  EMBED Equation.3  (ii) Using Poisson distribution P(X=2) =  EMBED Equation.3  Comment : There is a difference between the two probabilities because of the fact that Poisson distribution (P.D.) is an approximation to binomial distribution (B.D.) and it is applicable for large n. Hence the solution. Problem 5# A hospital switch board receives an average of 4 emergency calls in a 10 min. interval. What is the probability that (i) there are at the most 2 emergency calls and (ii) there are exactly 3 emergency calls in a 10 min. interval? Solution: Given (=4, (i) P(X( 2)=P(X= 0)+P(X=1)+ P(X= 2) =  EMBED Equation.3 = e-([1+(+ EMBED Equation.3 ] = e-4[1+4+8] = 13 e-4 = 0.238. (ii) P(X= 3) =  EMBED Equation.3  Hence the solution. Chapter 2 Probability Distributions Tutorial 12 Poissons By N V Nagendram --------------------------------------------------------------------------------------------------------------- Problem 6# A rent a car firm has two cars which it hires from day to day. The number of demands for a car on each day is distributed as a Poisson variate with mean 1.5. Calculate the proportion of days on which (i) neither car is used (ii) some demand is refused? Problem 7# In a Poisson distribution (P.D.), P(X = 0) = 2 P(X = 1), then find P(X = 2)? Problem 8# In a factory which turns out razor blades, there is a chance of 0.002 for any blade to be defective. The blades are supplied in packets of 10 each. Using Poisson distribution, Calculate the approximate number of packets containing no defective, one defective and two defective blades if there are 10,000 such packets? Problem 9# the probability of getting no misprint in a page of a book is e-4. Determine the probability that a page of a book contains more than 2 misprints? Problem 10# Obtain the Poisson distribution (P.D.) as a limiting case of Binomial distribution? Problem 11# Fit a Poisson distribution to the following data and calculate the theoretical frequencies: x01234y46382291Solution: Mean = E(X) = ( and Variance V(X) = (2 = E(X2) [E(X)]2 xififi xixi2fi xi2 046000138381382224448839279814141616 EMBED Equation.3  EMBED Equation.3  EMBED Equation.3 Mean =  EMBED Equation.3 ; Variance =  EMBED Equation.3  ( Mean =Variance = ( = 0.974. The theoretical frequencies are f(x) = N. P(X=x) f(0) = 116. P(X=0) = 116. E-0.974 = 44 f(1) = 116. P(X=1) =  EMBED Equation.3  f(2) = 116. P(X=2) =  EMBED Equation.3  f(3) = 116. P(X=3) =  EMBED Equation.3  f(4) = 116. P(X=4) = 116 {f(0) +f(1)+f(2)+f(3)} = 116 114 = 2 Hence the solution. Problem 12# If a bank receives on an average 6 bad cheques per day, what are the probabilities that it will receive (i) four bad cheques on any given day (ii) 10 bad cheques on any two consecutive days. Solution: Let  (t  T ( = np ( p =  EMBED Equation.3  ( = np =  EMBED Equation.3  P(X=x) = [e-(t ((T)x ]/x! ( = 6, T = 1 and ( ( = (T = 6 f(4,6) = e-6 . 64 = 0.1339 4! F(10; ()=  EMBED Equation.3  Hence the solution. Chapter 2 Probability Distributions Tutorial 13 Poissons Process By N V Nagendram --------------------------------------------------------------------------------------------------------------- Problem 1# Fit a Poisson distribution to the following x:01234y:46382291 Problem 2# Fit a Poisson distribution to the set of observations as below x:01234y:122601521 Problem 3# The incidence of occupational disease in an industry is such that the workmen have a 10% chance of suffering from it. What is probability of 7, five or more will suffer from it? Problem 4# A car hire firm has two cars which it hires out day by day. The number of demands for a car on each day is distributed as a Poisson distribution with mean 1.5. calculate the proportion of days. (i) on which there is no demand (ii) on which demand is refused (e-5 = 0.2231)? [Ans. i)0.2231 ii)0.1913] Problem 5# If a random variable has a Poisson distribution such that P(1) = P(2) find (i) mean of the distribution (ii) P(4) ? [Ans. i) 2 ii) (2/3).e- 2] Problem 6# If the probability of a bad reaction from a certain injection is 0.001, determine the chance that out of 2,000 individuals more than two will get a bad reaction?[Ans.0.32] Problem7 # If 3 % of the electric bulbs manufactured by a company are defective, find the probability that in a sample of 100 bulbs (i) 0 (ii) 1 (iii) 4 [Ans. i) 0.04979 ii)0.1494 iii) 0.1008] Problem 8# Ten present of the tools produced in a certain manufacturing process turn out to be defective. Find the probability that in a sample of 10 tools chosen at random exactly two will be defective by using the Poisson approximation to the binomial distribution?[Ans.0.18] Chapter 2 Probability Distributions Tutorial 14 Normal Distributions By N V Nagendram --------------------------------------------------------------------------------------------------------------- Problem 1# Show that the mean deviation from the mean for normal distribution (N.Dn) is equal to 4/5 of standard deviation approximately? [Ans. M.D=4/5(] Problem 2# X is normally distributed with mean 12 and S.D = 4then find (i) P(0(X(12) (ii) P(X ( 20) (iii) P(X ( 20) (iv) if P(X > C) = 0.24. [Ans. i)0.4896 ii)0.9772 iii) 0.0228 iv) 0.24 and C= 14.84] Problem 3# Show that the mean deviation from the mean for the normal distributon [N.Dn]is 4/5 of standard deviation approximately. [Ans. ( =0.79(=4/5(] Problem 4# Xis a normal variate with mean 30 and standard deviation 5. Find the probabilities that (i) 26 ( X ( 40 (ii) X ( 45. [Ans. i) 0.2882+0.4772=0.7653 ii) 0.0013] Problem 5# A random variable has normal distribution with ( = 62.4. find its standard deviation if the probability is 0.20 that it will take on a value greater than 79.2. [Ans. (=20] Problem 6# find the probabilities that a random variable having a standard normal distribution will take on a value (i) between 0.87 and 1.28 (ii) between 0.34 and 0.62. [Ans. i) 0.0919 ii) 0.1443 + 0.2343 = 0.3767] Problem 7# In a normal distribution (N.Dn) 31% of the items are under 45 and 8% are over 63. Find the mean and variance of the distribution. [Ans. (=50, (=10] Problem 8# In a normal distribution (N.Dn), 7% of the items are under 35 and 89% are over 64. Find the mean and variance of the distribution. [Ans. (=50.3, (=10.33] Chapter 2 Lecture 7 Sampling Sampling Distributions by N. V. Nagendram Sampling Distributions: Introduction The field of statistics deals with the collection presentation, analysis and use of data to make decision and solve problems. The main objective of any statistical study is to draw conclusions about a collection of objects under study. This collection is called the Population. Instead of examining this population, which may be difficult or impossible to do, one may arrive at the idea of examining only a small part of this population, which is called a sample. This can be done with the aim of drawing inferences about the population by using information from the sample, this process is known as statistical inference. The process of drawing samples is called sampling. A sample is a true or good representative of the population, if the sampling method is probabilistic. The most important of all probabilistic samplings is the random sampling, in which each member of the population has the equal chance of being included in the sample. Samples will be used to draw inferences about population, by estimating the parameters of population, such as mean () , standarad deviation (() etc., Estimation of population parameters is possible only by studying some relevant statistical quantities computed from a sample of the population called sample statistics (or) simply statistic is often used for the random variable or for its value, the particular sense being clear from the context. Let us consider all possible samples of a population and calculate a statistic for instance sample mean. Then the set of all such b\values, one for each sample, is called the sampling distribution of the statistic. Now we can compute the statistics mean variance etc., for this sampling distribution. In most statistic problems, it is necessary to use the information from sample to draw inferences about the population. Definition: Population The population in a statistical study is the set or collection or totality of observations about which inferences are to be drawn. Thus the population consists of sets of numbers, measurements or observations. Population size N is the number of objects or observations in the population. Population is said to be finite or infinite depending on the size N being finite or infinite. Since it is impracticable to examine the entire population, a finite subset of the population known as sample is studied. Sample size n is the number of objects or observations in the sample. Example: (i) Engineering graduate students in A.P. (Population), Engineering graduate students of a college (Sample) Population Sample   Example: Total production of items in a month (Population), Total production of items in one day (Sample) Example: Budget of India (Population), Budget of A.P. (Sample), budget of a district (sub sample) Population Sample   Sub sample  Definition: Population parameter: A statistical measure or constant obtained from the population is called population parameter. Example: population mean (), population variance ((2). Definition: (Sample) statistic: A statistical measurement computed from sample observations is called a (sample) statistic. Example: sample mean ( EMBED Equation.3 ), sample variance (s2) clearly, parameters are to population while statistics are to sample. , (, p represent the population mean, population standard deviation, population proportion, similarly  EMBED Equation.3 , s , p denote sample mean, sample standard deviation(s. s. d.), sample proportion. Note: The samples must be a true or good representative of the population, sampling should be random or probabilistic. Definition: Sampling: The process of drawing or obtaining samples is called sampling. Definition: Large sampling: If n e" 30, then the sampling is known as large sampling. Definition: Small sampling: If n < 30, then the sampling is known as small or exact sampling. Note: The simplest and most commonly used type of probabilistic sampling is the random sampling. Definition: Random Sampling: Each member of the population has equal chances or probability of being included in the sample. The sample obtained by this method is termed as a random sample. Definition: Finite Population: Population may be finite or infinite. If the number of items or observations consisting the population is fixed and limited, it is called as finite population. Example: The workers in a factory, student in a college etc.,     Definition: Infinite Population: If the number of items or observations consisting the population is infinite (not fixed and not limited), it is called as finite population. Example: The population of all real numbers lying between 0 and 1. The population of stars or astral bodies in the sky. Definition: Sampling with replacement: If the items are selected or drawn one by one such a way that an item drawn at a time is replaced back to the population before the next or subsequent draw, it is known as (random) sampling with replacement. In this type of sampling from a population of size N, the probability of a selection of a unit at each draw remains  EMBED Equation.3  . Thus sampling from finite population with replacement can be considered theoretically as sampling from infinite population. In this, Nn samples will be drawn. Definition: In Sampling without replacement: An item of the population cannot be chosen more than once, as it is not replaced. In this NCn samples will be drawn. Hence the probability of drawing a unit from a population of N items at r th draw is  EMBED Equation.3 . Statistic is a real-valued function of the random sample. So it is a function of one or more random variables not involving any unknown parameter. Thus statistic is a function of samples observations only and is itself a random variable. Hence a statistic must have a probability distribution. Definition: Sample mean: Let x1, x2, x3,. . . , xn be a random, sample of size n from a population. Then sample mean = ( EMBED Equation.3 ) =  EMBED Equation.3 . Definition: sample Variance: Then sample variance = s2 =  EMBED Equation.3 . Sample standard deviation is the positive square root of sample variance. Sample mean and sample variance are two important statistics which are statistical measures of a random sample of size n. Chapter 2 Lecture 8 Sampling Sampling Distributions by N. V. Nagendram Sampling Distribution: Let us consider all possible samples of size n, from a finite population of size N. Then the total number of all possible samples of size n, which can be drawn from the population is NCn = m. Compute a statistic ( [such as mean, variance /s.d, proportion] for each of these sample using the sample data x1, x2, x3,. . . , xn by ( = (( x1, x2, x3,. . . , xn) Sample number123. . .mStatistic ((1(2(3. . .(m Sampling distribution of the statistic ( is the set of values {(1, (2, (3, . . ., (m} of the statistic ( Obtained, one for each sample. Thus sampling distribution describes how a statistic ( will vary from one sample to the other of the same size. Although all the m samples are drawn from the given population, the items included in different samples are different. If the statistic ( is mean, then the corresponding distribution of the statistic is known as sampling distribution of means, thus if ( is variance, proportion etc., the corresponding distribution is known as sampling distribution of variances, sampling distribution of proportions etc., Then Mean of sampling distribution of ( = ( EMBED Equation.3 ) =  EMBED Equation.3 . And Variance of sampling distribution of ( =  EMBED Equation.3 . Similarly we can have mean of sampling distribution of means, variance of sampling distribution of means, variance of the sampling distribution of variances etc., Standarad Error: The standard deviation of the sampling distribution of a statistic is known as standard error (SE). The standard error gives some idea about the precision of the estimate of the parameters. As the sample size n increases, S.E. decreases. S.E. plays a very important role in large sample decision theory and forms the basis in hypothesis testing. Sampling distribution of a statistic enables us to know information about the corresponding population parameter. Degrees of freedom ((): The number of degrees of freedom usually denoted by greek alphabet (, is a positive integer equals to n k where n is the number of independent observations of the random sample and k is the number of population parameters which are calculated using the sample data. The degrees of freedom ( = n - k is the difference between n the sample size and k the number of independent contains imposed on the observations in the sample. The sampling distribution of the Mean (( known): To answer any questions related to sampling distribution of the mean ( EMBED Equation.3 ) we need to consider a random sample of n observations and determine the value  EMBED Equation.3  for each sample, then by various values of  EMBED Equation.3 , it may be possible to get an idea of the nature of the sampling distribution. Aslo we have to consider the following theorem for the mean ( EMBED Equation.3  and the variance ( EMBED Equation.3 of sampling distribution of the mean ( EMBED Equation.3 ). Theorem: If a random sample of size n is taken from a population having the mean ( and the variance (2 , then ( EMBED Equation.3 ) is a random variable whose distribution has the mean (. Proof: For samples from infinite population the variance of this distribution is  EMBED Equation.3 . For samples from finite population the variance of this distribution is  EMBED Equation.3  By above statement, population is infinite then sampling with replacement ( EMBED Equation.3  = ( and ( EMBED Equation.3 =  EMBED Equation.3  And when the population is finite, size N (sampling without replacement) ( EMBED Equation.3  = ( and ( EMBED Equation.3 =  EMBED Equation.3  EMBED Equation.3  Note: The factor  EMBED Equation.3 is known as finite population correction factor. In sampling with replacement, we will have Nn samples each with probability  EMBED Equation.3  In sampling without replacement we will have NCn samples each with probability  EMBED Equation.3 . Note: The factor  EMBED Equation.3 can be neglected if N is too large compared to the sample size n. Chapter 2 Lecture 9 Sampling Sampling Distributions by N. V. Nagendram Central limit theorem: Whenever n is large, the sampling distribution of  EMBED Equation.3 approximately normal with mean ( and variance  EMBED Equation.3  regardless of the form of the parent population distribution, as the following theorem states [without proof] Theorem: If  EMBED Equation.3  is the mean of a random sample of size n drawn from a population with mean ( and finite variance (2 then the standardized sample mean Z =  EMBED Equation.3  is a random variable whose distribution function approaches that of the standard normal distribution N(0, 1) as n ( (. Normal distribution provides a good approximation to the sampling distribution for almost all the populations for n ( 30. For n < 30 small samples, sampling distribution of  EMBED Equation.3  is normally distributed, provided sampling is from normal population. Sampling distribution of proportions: Suppose that a population is infinite and that the probability of occurance of an event called its success is p, while the probability of non-occurance of the event is q = 1 p. Consider all possible samples of size N drawn from tis population, and for each sample compute the proportion p of successes. Then, we can have a sampling distribution of proportions whose mean (p and standard deviation (p are given by (p = p and (p2 =  EMBED Equation.3  .(1) While population is binomially distributed, the sampling distribution of proportion is normally distributed whenever n is large ( 30. Equation (1) are also valied for a finite population in which sampling is with replacement. For finite population sampling without replacement of size N (p = p and (p2 =  EMBED Equation.3  Sampling distributions of differences and sums: Let (S EMBED Equation.3  and ( S EMBED Equation.3  be the mean and standard deviation of a sampling distribution of statistic S1 obtained by calculating S1 for all possible samples of size n1 drawn from population 1. This yields a sampling distribution of the statistic S1. In a similar manner, (S EMBED Equation.3  and ( S EMBED Equation.3 be the mean and standard deviation of sampling distribution of statistic S2 obtained by calculating S2 for all possible samples of size n2 drawn from another different population 2. Now we can have a distribution of differences S1 S2, called the sampling distribution of differences of the statistics, from the two population 1 and 2. Then the mean (S EMBED Equation.3 - S EMBED Equation.3  and the standard deviation (S EMBED Equation.3 - S EMBED Equation.3 the sampling distribution of differences are given by (S EMBED Equation.3 - S EMBED Equation.3 = (S1 (S2 and (S EMBED Equation.3 - S EMBED Equation.3 = EMBED Equation.3  provided the samples are independent. Sampling distribution of sum of statistics has mean (S EMBED Equation.3 + S EMBED Equation.3 = (S1 + (S2 and (S EMBED Equation.3 + S EMBED Equation.3 = EMBED Equation.3  provided the samples are independent. For infinite population the sampling distribution of the differences of means has mean (( EMBED Equation.3  )and (( EMBED Equation.3 ) given by (( EMBED Equation.3  ) = (( EMBED Equation.3 -  EMBED Equation.3  = ( EMBED Equation.3 - ( EMBED Equation.3  and (( EMBED Equation.3  ) =  EMBED Equation.3  =  EMBED Equation.3 . For infinite population the sampling distribution of sums of means has mean (( EMBED Equation.3  )and (( EMBED Equation.3 ) given by (( EMBED Equation.3  ) = (( EMBED Equation.3 + EMBED Equation.3  = ( EMBED Equation.3 + ( EMBED Equation.3  and (( EMBED Equation.3  ) =  EMBED Equation.3  =  EMBED Equation.3 . Sampling distribution of mean ( unknown: t-distribution: To estimate or infer on a population mean or the difference between two population means, it was assumed that the population standard deviation ( is known. When ( is unknown, for large n ( 30, ( can be replaced by the sample standard deviation s, calculated using the sample mean  EMBED Equation.3  by the formula = s2 =  EMBED Equation.3 . For small sample of size n < 30 the unknown ( can be substituted by s, provided we make an assumption that the sample is drawn from a normal population. A random variable having the t-distribution: Let  EMBED Equation.3 be the mean of a random sample of size n drawn from a normal population with mean ( and variance (2 then t =  EMBED Equation.3  is a random variable having the t-distribution with ( = n 1 degrees of freedom. Where s2 =  EMBED Equation.3 . This result is more general than previous theorem CLT in the sense that it does not require knowledge of (: on the other hand, it is less general than the previous theorem CLT in the sense that it requires the assumption of normal population. Thus for all small samples n < 30 and with ( unknown a statistic for inference on population mean ( is t =  EMBED Equation.3  With the underlying assumption of sampling from normal population. The t-distribution curve is symmetric about the mean 0, bell shaped and asymptotic on both sides of horizontal t-axis. Thus t-distribution curve is similar to normal curve. The variance for the t-distribution is more than 1 as it depends on the parameter ( = n 1 degrees of freedom.   but it approaches 1 as n ( (. In essence, as ( = (n 1 ) ( (, t-distribution tends to the standard normal distribution. Clearly for n( 30, standard normal distribution provides a good approximation to the t-distribution. Critical values of t-distribution is denote by t(, which is such that the area under the curve to the right of t( equals to (. Since the t-distribution is symmetric, it follows that t 1 - ( = - t( i.e., the t-value leaving an area of 1 - ( to the right and therefore an area ( to its left, is equal to the negative t-value which leaves an area ( in the right tail of the distribution.   Please observe critical values of t( for values of the parameter (. In tables the left-hand column contains values of (, the column headings are area ( in the right hand tail of the t-distribution, the entries are values of t(. Chapter 2 Lecture 10 (2- Distribution Sampling Distributions by N. V. Nagendram Definition: (2 (chi squared) distribution is a continuous probability distribution of a c.r.v. X with probability density function given by  EMBED Equation.3  Where ( is a +ve integer is the only single parameter of the distribution, also known as degrees of freedom. Properties of (2- Distribution: (i) (2- Distribution curve is not symmetrical, lies entirely in the first quadrant. And hence not a normal curve, since (2 varies from 0 to (. (ii) It depends only on the degrees of freedom (. (iii) If X12 and X22 are two independent distributions with (1, (2 degrees of freedom then (12+(22 will be chi- squared distributions with ((1 + (2) degrees of freedom i.e, it is additive.   Hence ( denotes the area under the chi-squared distribution to the right of ((2. So ((2 represents the (2-value such that the area under the (2-curve to its right is equal to (. (2-distribution is very important in estimation and hypothesis testing (2-distribution is used in sampling distributions, analysis of variance mainly, it is used as a measure of goodness of fit and in analysis of r ( c tables. For various values of ( and (, the values of ((2 are tabulated. In (2- table the left-hand column contains values of ( (degrees of freedom), the column headings are areas ( in the right hand tail of (2-distribution curve, the entries are (2- values. It is necessary to calculate values of ((2 for ( > 0.50, since ((2 curve or distribution is not symmetrical. Sampling distribution of Variance s2: From the earlier discussions, the sample mean is used to estimate the population mean. Similarly, the sample variance is used to estimate the population variance ((2). The sample variance is usually denoted by s2 and is given by  EMBED Equation.3 . A random variable having the (2-distribution: Theorem: If s2 is the variance of a random sample of size n from a normal population having the variance (2 then (2 =  EMBED Equation.3  is a random variable having the (2-distribution with ( = n 1 dof. Exactly 95% of (2-distribution lies between (20.975 and (20.025 when (2 is too small. (2-value falls to the right of (20.025 and when (2 is too large, (2 falls to the left of (20.975. thus when (2 is correct (2-value fall s to the left of (20.975 or to the right of (20.025. critical region for testing : H0: (2 = (02 Alternate hpothesisReject H0 if(2 < (02(2 ( (21-((2 > (02(2 ( ((2(2 ( (02(2 ( (2(1-()/2 F-Distribution (sampling distribution of the ratio of two sample variances): Definition: Another important continuous probability distribution which plays an important role in connection with sampling from normal population is the F-distribution. If s12 and s22 are the variances of independent random samples of size n1 and n2 from normal populations with variances (12 and (22. To determine whether the two samples come from two populations having equal variances, consider the sampling distribution of the ratio of the variances of the two independent random samples defined by  EMBED Equation.3  which follows F-distribution with (1 = n1 1 and (2 = n2 1 degrees of freedom. Uses: F-distribution can be used for testing the quality of several population means, comparing sample variances, and analysis of variance completely depends on F-distribution. Under the hypothesis that two normal populations have the same variance : (12 = (22, we have  EMBED Equation.3 . F determines whether the ratio of two sample variances s1 and s2 is too small or too large. When F is close to 1, the two sample variances s1 and s2 are almost same. F is always a positive number whenever the larger sample variance as the numerator.  f(F) f(F) (1 = 5, (2 = 5   (1 = 5, (2 = 15    0 1 2 3 4 5 6 10 F0.05 F0.01 Probability density functions of several F distributions Tabulated values of F Properties of F-distribution: (i) F-distribution curve lies entirely in first quadrant. (ii) The F-curve depends not only on the two parameters (1 and (2 but also on the order in Which they are stated. (iii) F1 -(((1, (2) =  EMBED Equation.3  where  EMBED Equation.3  is the value of F with (1 and (2 degrees of Freedom such that the area under the F-distribution curve to the right of F( is (. Note: Critical regions for testing the null hypothesis: (12 = (22 Alternate hypothesisTest statisticReject H0 if:(12 < (22 EMBED Equation.3 F> F((n2 1, n1 1)(12 > (22 EMBED Equation.3 F> F((n1 1, n2 1)(12 ( (22 EMBED Equation.3 F> F(/2(nM 1, nm 1) Chapter 2 Probability Distributions Tutorial 15 Sampling - Population PROBLEMS by N V Nagendram --------------------------------------------------------------------------------------------------------------- Problem 1# Find the value of the finite population correction factor for (i) n = 10 and N = 1000 (ii) n = 100 and N = 1000 ? Problem 2# A random sample of size 2 is drawn from the population 3,4,5. Find (i) population mean (ii) Population S.D. (iii) Sampling distribution (SD) of means (iv) the mean of SD of means (v) S.D of SD means? Problem 3# A random sample of size 2 is drawn from the population 3,4,5. Find (i) population mean (ii) Population S.D. (iii) Sampling distribution (SD) of means (iv) the mean of SD of means (v) S.D of SD means? Solve the problem without replacement? [Ans.0.4082] Problem 4# Determine the mean and s.d of sampling distributions of variances for the population 3,7,11,15 with n = 2 and with sampling (i) with replacement and (ii) without replacement? [Ans. 11.489] Problem 5# Find P EMBED Equation.3  if a random sample size 36 is drawn from an infinite population with mean ( = 63 and s.d. ( = 9. [Ans. 0.0062] Problem 6# Determine the probability that mean breaking strength of cables produced by company 2 will be (i) at least 600N more than (ii) at least 450 N more than the cables produced by company 1, if 100 cables of brand 1 and 50 cables of brand 2 are tested. companyMean breaking strengths.d.Sample size14000 N300 N10024500 N200 N50 [Ans. 0.8869] Problem 7# Let  EMBED Equation.3  and  EMBED Equation.3 be the average drying time of two types of oil paints 1 and 2 for samples size n1 = n2 = 18. Suppose (1 = (2 = 1. Find the value of P( EMBED Equation.3  -  EMBED Equation.3  > 1), assuming that mean drying time is equal for the two types of oil paints. [Ans. 0.0013] Problem 8# A company claims that the mean life time of tube lights is 500 hours. Is the claim of the company tenable if a random sample of 25 tube lights produced by th company has mean 518 hours and s.d. 40 hours. [Ans. 2.492] Problem 9# Determine the probability that the variance of the first sample of size n1 = 9 will be at least 4 times as large as the variance of the second sample of size n2 = 16 if the two samples are independent random samples from a normal population. [Ans. 0.01] Problem 10# Is there reason to believe that the life expected of group A and Group B is same or not from the following data GroupA3439.246.148.749.445.955.342.743.756.6Group B49.755.457.054.250.444.253.457.561.958.2 [Ans. 1.63] Problem 11# A random sample of size 25 from a normal population has the mean  EMBED Equation.3 =47.5 and the standard deviation s = 8.4. does this information tend to support of refute the claim that the mean of the population is ( = 42.1? [Ans. t =3.21] Problem 12# In 16 hour ten runs, the gasoline consumption of an engine averaged 16.4 gallons with a. s. d. of 2.1 gallons. Test the claim that the average gasoline consumption of this engine is 12.0 gallons per hour. [Ans. t =8.38] Problem 13# Suppose that the thickness of a part used in a semiconductor is its critical dimension, and that process of manufacturing these parts is considered to be under control if the true version among the thickness of the parts is given by a standard deviation not greater than ( = 0.60 thousandth of an inch. To keep a check on the process, random samples of size n = 20 are taken periodically, and is regarded to be out of control if the probability that s2 will take on a value greater than or equal to the observed sample value is 0.01 or less even though ( = 0.60 what can one conclude about the process if the standard deviation of such a periodic random sample is s = 0.84 thousandth of an inch? [Ans.37.24] Problem 14# A soft-drink vending machine is set so that the amount of drink dispensed is a random variable with a mean of 200 millilitres and a standard deviation of 15 millilitres. What is the probability that the average (mean) amount dispensed in a random sample size of 36 at least 204 millilitres? Problem 15# If two independent random sample of size n1 = 7 and n2 = 13 are taken from a normal population what is the probability that the variance of the first sample will be at least three times as large that of the second sample? Problem 16# The claim that the variance of a normal population is (2 = 21.3 is rejected if the variance of a random sample of size 15 exceeds 39.74. What is the probability that the claim will be rejected even though (2 = 21.3? [Ans.0025] Problem 17# An electronic company manufactures resistors that have a mean resistance of 100 ( and a standard deviation of 10 (. The distribution of resistance is normal. Find the probability that a random sample 25 resistors will have an average resistance less than 95 (? [Ans. 0.0062] Problem 18# The mean voltage of a battery is 15 volt and s.d.is 0.2 volt. What is the probability that four such batteries connected in series will have a combined voltage of 60.8 or more volts? [Ans. 0.0228] Problem 19# Certain ball bearings have a mean weight of 5.02 ounces and standard deviation of 0.30 ounces. Find the probability that a random sample of 100 ball bearings will have a combined weight between 496 and 500 ounces? [Ans. 0.2318] Problem 20# A manufacturer of fuses claims that with a 20% overload, the fuses will blow in 12.40 minutes on the average. To test the claim, a sample of 20 of the fuses was subjected to a 20% overload, and the times it took them to blow had a mean of 10.63 minutes and a s.d. of 2.48 minutes. If it can be assumed that the data constitute a random sample from a normal population, do they tend to support or refute the manufacturers claim? [Ans.- 3.19] Problem 21# show that for random samples of size n from a normal population with the variance (2, the sampling distribution of (2 has the mean (2 and the variance  EMBED Equation.3 ? Problem 22# If S12 and S22 are the variances of independent random samples of size n1 = 10 and n2 = 15 from normal population with equal variances find P(S12/ S22 < 4.03)?[Ans. 0.99] Problem 23# A random sample of size n = 25 from a normal population has the mean  EMBED Equation.3  = 47 and the standard deviation ( = 7. It we base our decision on the statistic, can we say that the given information supports the conjecture that the mean of the population is ( = 42? Problem 24# The claim that the variance of a normal population is (2 =4 is to be rejected if the variance of a random sample of size 9 exceeds 7.7535. What is the probability that this claim will be rejected even though (2 =4? [Ans. 0.5] Problem 25# A random sample of size n = 12 from a normal population  EMBED Equation.3  = 27.8 has the mean and the variance (2 = 3.24. it we base our decision on the statistic can we say that the given information supports the claim that the mean of the population is ( = 28.5?[Ans.-1.347] Problem 26# The distribution of annual earnings of all bank letters with five years experience is skewed negatively. This distribution has a mean of Rs.19000 and a standard deviation of Rs.2000. If we draw a random sample of 30 tellers, what is the probability that the earnings will average more than Rs.19750 annually? [Ans. 0.0202] Problem 27# If a gallon can of paint covers on the average 513.3 square feet(Ft2.) with a standard deviation(s.d.) of 31.5 square feet(Ft2.). what is the probability that the mean area covered by a sample of 40 of these 1 gallon cans will be anywhere from 510 to 520 square feet(Ft2.)? [Ans.0.6553] Problem 28# A random sample of 100 is taken from an infinite population having the mean ( = 76 and the variance = (2 = 256. Find the probability that  EMBED Equation.3  will be between 75 and 78? [Ans. 0.6268] Problem 29# If two independent random samples of size n1 = 13 and n2 = 7 are taken from a normal population. What is the probability that the variance of the first sample will be atleast four times as that of the second sample? [Ans. 4.00] Problem 30# If two independent random samples of size n1 = 26 and n2 = 8 are taken from a normal population. What is the probability that the variance of the second sample will be atleast 2.4 times as that of the first sample? [Ans. 0.05] Problem 31# If the actual amount of instant coffee which a filing machine puts into 6-ounce jars is r. v. having a normal distribution with s.d. 0.05 ounce and if only 3% of the jars are to contain less than 6 ounces of coffee, what must be the mean fill of these jars? [Ans. (=6.094] Problem 32# A manufacturer of a certain type of synthetic fishing line has found from long experience of testing that the breaking strength of his product has an approximate normal distribution with a mean of 30 pounds( lbs. ) and a standard deviation of 4 pounds( lbs. ). A time and money saving change in the manufacture process of the product is tried. A sample of 25 testing length pieces of the new process line is taken and tested with a resulting sample mean of 28 pounds(lbs.) What is the probability of obtaining a mean as low as 28 if the process has had no harmful effect on breaking strength? [Ans. 0.006] Problem 33# An Urn contains 1000 white and 2000 black balls. If X denotes the number of white balls when 300 balls are drawn without replacement, then find P(180 < X < 120)? [Ans. 0.9858] Problem 34# Two movie theatres compete for 900 visitors. Suppose each visitor chooses one of the two balls independent of the choice of the other visitors; how many seats should each theatre have so that the probability of turning away any visitor for lack of seats is less than 1%? [Ans. 489] Problem 35# Let X be a random variable where (x is unknown as (x2 = 0.25 i.e.,1/4 Find out how large a random sample must be taken in order that the probability will be at test 0.95 and the sample mean  EMBED Equation.3  will lies within 0.25 of the population mean? [Ans. 80] Problem 36# If a random sample of size n is selected from the finite population that consists of the integers 1,2,3,. . . ,N show that (i) the mean  EMBED Equation.3  is  EMBED Equation.3  (ii) the variance of  EMBED Equation.3  is  EMBED Equation.3  (iii) the mean and the variance of Y = n.  EMBED Equation.3  are E(Y) =  EMBED Equation.3  and the var(Y) =  EMBED Equation.3 ? Problem 37# How many different samples of size n =3 can be drawn from a finite population of size (a) N =12 (b) N = 20 (c) N = 50 [Ans. a) 220, b) 1140 c) 19600] Problem 38# What is the probability of each possible sample if (i) a random sample of size n =4 is to be drawn from a finite population of size N = 12 (ii) a random sample of size n = 5 is to be drawn from a finite population of size N = 22? [Ans. a) 1/495 b) 1/77] Problem 39# Independent random samples of size n1 = 30 and n2 = 50 are taken from two normal populations having the means (1 = 78 and (2 = 78 and the variances (12 and (22. Find the probability that the mean of the first sample will exceed that of the second sample by at least 4.8? [Ans. 0.2743] Problem 40# If S1 and S2 are the variances of independent random samples of size n1 = 61 and n2 = 31 from normal population with (12 = 12 and (22 = 18 Find  EMBED Equation.3  [Ans. 0.05] Chapter 2 Probability Distributions Tutorial 15 Sampling - Population by N V Nagendram --------------------------------------------------------------------------------------------------------------- Problem 1# Find the value of the finite population correction factor for (i) n = 10 and N = 1000 (ii) n = 100 and N = 1000 ? Solution: (i)  EMBED Equation.3  (ii)  EMBED Equation.3  Hence the solution. Problem 2# A random sample of size 2 is drawn from the population 3,4,5. Find (i) population mean (ii) Population S.D. (iii) Sampling distribution (SD) of means (iv) the mean of SD of means (v) S.D of SD means? Solution: (i) Population mean = ( =  EMBED Equation.3  (ii) s.d. of population = ( =  EMBED Equation.3  (iii) sampling with replacement (infinite population): The total number of samples with replacement is Nn = 32= 9 here N = population size and n = sample size. Listing all possible samples of size 2 from population 3,4,5 with replacement, we get 9 samples as below:  EMBED Equation.3  Now compute the statistic the arithmetic mean for each of these 9 samples the set of 9 samples means  EMBED Equation.3 , gives rise to the distribution of means of the sample known as sampling distribution of means 3 3.5 4 3.5 4 4.5 4 4.5 5 This sampling distribution of means can also be arranged in the form of frequency distribution Sample mean  EMBED Equation.3 i 33.544.55Frequency fi12321 (iv) Mean of the sampling distribution of means = ( EMBED Equation.3 = EMBED Equation.3  Showing ( EMBED Equation.3 =(= 4 (v) (2 EMBED Equation.3 =  EMBED Equation.3  therefore ( EMBED Equation.3 = 0.5773 Problem 3# A random sample of size 2 is drawn from the population 3,4,5. Find (i) population mean (ii) Population S.D. (iii) Sampling distribution (SD) of means (iv) the mean of SD of means (v) S.D of SD means? Solve the problem without replacement? [Ans.0.4082] Solution: = 4 (ii) ( = 0.8164 (iii) Sampling without replacement finite population the toal number of samples without replacement is Ncn = 3C2 = 3 the three saples are (3,4), (3,5) (4,5) and their means are 3.5, 4. 4.5 (iv) ( EMBED Equation.3 == mean of smpling distribution of means =  EMBED Equation.3 =( (2 EMBED Equation.3 =  EMBED Equation.3  ( EMBED Equation.3 = 0.4082. Hence the solution. Problem 4# Determine the mean and s.d of sampling distributions of variances for the population 3,7,11,15 with n = 2 and with sampling (i) with replacement and (ii) without replacement? [Ans. 11.489] Solution: (i) Nn = 42 = 16 samples (3,3),(3,7) , . . ., (15,11), (15,15) With Means3579111315Frequency1234321Variances041636  EMBED Equation.3 = 10; (2S2 =  EMBED Equation.3 =11.489 Hence the solution. Problem 5# Find P EMBED Equation.3  if a random sample size 36 is drawn from an infinite population with mean ( = 63 and s.d. ( = 9. [Ans. 0.0062] Solution: let z =  EMBED Equation.3  Hence P EMBED Equation.3 = P(Z> 2.50) = 0.0062. Hence the solution. Problem 6# Determine the probability that mean breaking strength of cables produced by company 2 will be (i) at least 600N more than (ii) at least 450 N more than the cables produced by company 1, if 100 cables of brand 1 and 50 cables of brand 2 are tested. companyMean breaking strengths.d.Sample size14000 N300 N10024500 N200 N50 [Ans. 0.8869] Solution: (( EMBED Equation.3 -  EMBED Equation.3 )=(( EMBED Equation.3 )- (( EMBED Equation.3 )= 4500 4000 = 500 N (( EMBED Equation.3 -  EMBED Equation.3 )= EMBED Equation.3  EMBED Equation.3  (i) P( EMBED Equation.3 -  EMBED Equation.3 > 600) = P(Z >  EMBED Equation.3 ) = P(Z > 2.4254) = 0.0078 (ii) P( EMBED Equation.3 -  EMBED Equation.3 > 450) = P(Z >  EMBED Equation.3 ) = P(Z > -1.2127) = 0.8869. Hence the solution. Problem 7# Let  EMBED Equation.3  and  EMBED Equation.3 be the average drying time of two types of oil paints 1 and 2 for samples size n1 = n2 = 18. Suppose (1 = (2 = 1. Find the value of P( EMBED Equation.3  -  EMBED Equation.3  > 1), assuming that mean drying time is equal for the two types of oil paints. [Ans. 0.0013] Solution: (2 ( EMBED Equation.3 -  EMBED Equation.3 )= EMBED Equation.3  P( EMBED Equation.3 -  EMBED Equation.3 ) = P(Z >  EMBED Equation.3 ) = P(Z >  EMBED Equation.3 = P(Z > 3) = 1- 0.9987 = 0.0013 Hence the solution. Problem 8# A company claims that the mean life time of tube lights is 500 hours. Is the claim of the company tenable if a random sample of 25 tube lights produced by th company has mean 518 hours and s.d. 40 hours. [Ans. 2.492] Solution: Given  EMBED Equation.3 = 518 hrs. n = 25, s = 40, ( = 500 t =  EMBED Equation.3  since, t = 2.25 < t0.01, v =24 = 2.492 Accept the claim of the company. Hence the solution. Problem 9# Determine the probability that the variance of the first sample of size n1 = 9 will be at least 4 times as large as the variance of the second sample of size n2 = 16 if the two samples are independent random samples from a normal population. [Ans. 0.01] Solution: From table F0.01 = 4 for (1 = n1 1= 9 1 (2 = n2 1 = 16 1 = 15, the desired probability is 0.01 [from F0.01 tables] Hence the solution. Problem 10# Is there reason to believe that the life expected of group A and Group B is same or not from the following data GroupA3439.246.148.749.445.955.342.743.756.6Group B49.755.457.054.250.444.253.457.561.958.2 [Ans. 1.63] Solution: Given data S2A =  EMBED Equation.3  S2B =  EMBED Equation.3  F =  EMBED Equation.3  clearly, variances empectancy is same for Group A and Group B. Hence the solution. Problem 11# A random sample of size 25 from a normal population has the mean  EMBED Equation.3 =47.5 and the standard deviation s = 8.4. does this information tend to support of refute the claim that the mean of the population is ( = 42.1? [Ans. t =3.21] Solution: given n =25,  EMBED Equation.3 =47.5, ( = 42.1, s = 8.4 we have from t-distribution t =  EMBED Equation.3 . This value of t has 24 degrees of freedom. From the table of t-distribution for ( = 24, we get probability that t will exceed 2.797 is 0.005. Then the probability of getting a value greater than 3.21 is negligible. Hence we conclude that the information given in the data of this example tend to refute the claim that the mean of the population is ( = 42.1. Hence the solution. Problem 12# In 16 hour ten runs, the gasoline consumption of an engine averaged 16.4 gallons with a. s. d. of 2.1 gallons. Test the claim that the average gasoline consumption of this engine is 12.0 gallons per hour. [Ans. t =8.38] Solution: substituting n = 16, (=12.0,  EMBED Equation.3 = 16.4 and s = 21 into the formula for t= EMBED Equation.3 , but from the table for ( = 15 the probability of getting a value of t greater than 2.947 is 0.005. the probability of getting a value greater than 8 must be negligible. Thus, it would seem reasonable to conclude that the true average hourly gasoline consumption of the engine exceeds 12.0 gasoline. Hence the solution. Problem 13# Suppose that the thickness of a part used in a semiconductor is its critical dimension, and that process of manufacturing these parts is considered to be under control if the true version among the thickness of the parts is given by a standard deviation not greater than ( = 0.60 thousandth of an inch. To keep a check on the process, random samples of size n = 20 are taken periodically, and is regarded to be out of control if the probability that s2 will take on a value greater than or equal to the observed sample value is 0.01 or less even though ( = 0.60 what can one conclude about the process if the standard deviation of such a periodic random sample is s = 0.84 thousandth of an inch? [Ans.37.24] Solution: The process will be declared out of control if  EMBED Equation.3  with n = 20 and ( = 0.60 exceeds (20.01,19 = 36.91, since  EMBED Equation.3 = 37.24 exceeds 36.191, the process is declared out of control. Of course it is assumed here that the sample may be regarded as a random sample from a normal population. Hence the solution. Problem 14# A soft-drink vending machine is set so that the amount of drink dispensed is a random variable with a mean of 200 millilitres and a standard deviation of 15 millilitres. What is the probability that the average (mean) amount dispensed in a random sample size of 36 at least 204 millilitres? Solution: The distribution of  EMBED Equation.3 has the mean (( EMBED Equation.3 ) = 200 and the standard deviation (( EMBED Equation.3 )= EMBED Equation.3 , and according to the central limit theorem, this distribution is approximately normal. And Z=  EMBED Equation.3 . Then P( EMBED Equation.3 ( 204) = P(Z ( 1.6) = 0.5000 0.4452 = 0.0548 Hence the solution. Problem 15# If two independent random sample of size n1 = 7 and n2 = 13 are taken from a normal population what is the probability that the variance of the first sample will be at least three times as large that of the second sample? Solution: F0.05((1 = 6, (2 =12) = 3 thus the desired probability is 0.05. Hence the solution. Problem 16# The claim that the variance of a normal population is (2 = 21.3 is rejected if the variance of a random sample of size 15 exceeds 39.74. What is the probability that the claim will be rejected even though (2 = 21.3? [Ans.0025] Solution: n = 15, (2 = 21.3, s2 = 39.74 (2 =  EMBED Equation.3  And (20.025, 14 = 26.119 (2 > (2  EMBED Equation.3  Therefore, probability that the claim will be rejected is 0.0025. Hence the solution. Problem 17# An electronic company manufactures resistors that have a mean resistance of 100 ( and a standard deviation of 10 (. The distribution of resistance is normal. Find the probability that a random sample 25 resistors will have an average resistance less than 95 (? [Ans. 0.0062] Solution: n = 25, (=100 (, ( = 10 ( so (( EMBED Equation.3 ) = 100 and (( EMBED Equation.3 ) = EMBED Equation.3  For  EMBED Equation.3  = 95, z =  EMBED Equation.3  Hence P( EMBED Equation.3  < 95) = P(Z < -2.5) = F(-2.5) = 1- F(2.5) = 1 0.9938 = 0.0062 Hence he solution. Problem 18# The mean voltage of a battery is 15 volt and s.d.is 0.2 volt. What is the probability that four such batteries connected in series will have a combined voltage of 60.8 or more volts? [Ans. 0.0228] Solution: Let, mean voltage of a batteries 1,2,3,4 be  EMBED Equation.3 , EMBED Equation.3 , EMBED Equation.3 , EMBED Equation.3  the mean of the series of the four batteries connected is (( EMBED Equation.3 + EMBED Equation.3 + EMBED Equation.3 + EMBED Equation.3  )= (( EMBED Equation.3 )+(( EMBED Equation.3 )+(( EMBED Equation.3 )+(( EMBED Equation.3 ) = 15 + 15 + 15 + 15 = 60 (( EMBED Equation.3 + EMBED Equation.3 + EMBED Equation.3 + EMBED Equation.3  )=  EMBED Equation.3  =  EMBED Equation.3  Let X be the combined voltage of the series. When x = 60.8, z =  EMBED Equation.3  Then the probability that the combined voltage is more than 60.8 is given by P(X ( 60.8) = P(Z ( 2) = 0.0228. Hence the solution. Problem 19# Certain ball bearings have a mean weight of 5.02 ounces and standard deviation of 0.30 ounces. Find the probability that a random sample of 100 ball bearings will have a combined weight between 496 and 500 ounces? [Ans. 0.2318] Solution: ( = 5.02, ( = 0.30, n = 100 (( EMBED Equation.3 ) = ( = 5.02 , ( ( EMBED Equation.3 ) =  EMBED Equation.3  P(4.96 <  EMBED Equation.3  < 0.5) = P EMBED Equation.3  = F(- 0.66) F(- 2) = F(2) F(0.66) = 0.9772 0.7454 = 0.2318 Hence the solution. Problem 20# A manufacturer of fuses claims that with a 20% overload, the fuses will blow in 12.40 minutes on the average. To test the claim, a sample of 20 of the fuses was subjected to a 20% overload, and the times it took them to blow had a mean of 10.63 minutes and a s.d. of 2.48 minutes. If it can be assumed that the data constitute a random sample from a normal population, do they tend to support or refute the manufacturers claim? [Ans.- 3.19] Solution: n = 20, (=12.40,  EMBED Equation.3  = 10.63, s = 2.48 then t =  EMBED Equation.3  Date refutes the producers claim since t = - 3.19 < - 2.861 with probability ( = 0.005. Hence the solution. Problem 21# show that for random samples of size n from a normal population with the variance (2, the sampling distribution of (2 has the mean (2 and the variance  EMBED Equation.3 ? Solution: We have  EMBED Equation.3  (  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  Hence the solution. Problem 22# If S12 and S22 are the variances of independent random samples of size n1 = 10 and n2 = 15 from normal population with equal variances find P(S12/ S22 < 4.03)?[Ans. 0.99] Solution: Let  EMBED Equation.3 and P EMBED Equation.3 = 1- P(F > 4.03) with 9 and 14 d.o.f. From table F0.01, 9.14  = 4.03 then the probability = 1 0.01 = 0.99 Hence the solution. Problem 23# A random sample of size n = 25 from a normal population has the mean  EMBED Equation.3  = 47 and the standard deviation ( = 7. It we base our decision on the statistic, can we say that the given information supports the conjecture that the mean of the population is ( = 42? Solution: f =  EMBED Equation.3  since, 3.57 exceeds t0.005, 24 = 2.797 for ( = 24 Clearly that the result is highly unlikely and conjecture is probably false. Hence the solution. Problem 24# The claim that the variance of a normal population is (2 =4 is to be rejected if the variance of a random sample of size 9 exceeds 7.7535. What is the probability that this claim will be rejected even though (2 =4? [Ans. 0.5] Solution: given (2 =4, n = 9, y =  EMBED Equation.3  P(y ( 2 (7.7535) = P(y ( 15.507) with 8 d.o.f. = 0.5 (table () Hence the solution. Problem 25# A random sample of size n = 12 from a normal population  EMBED Equation.3  = 27.8 has the mean and the variance (2 = 3.24. it we base our decision on the statistic can we say that the given information supports the claim that the mean of the population is ( = 28.5?[Ans.-1.347] Solution: The statistic is  EMBED Equation.3  since this is fairly small and close to t0, 10.11 the data tend to support the claim. Hence the solution. Problem 26# The distribution of annual earnings of all bank letters with five years experience is skewed negatively. This distribution has a mean of Rs.19000 and a standard deviation of Rs.2000. If we draw a random sample of 30 tellers, what is the probability that the earnings will average more than Rs.19750 annually? [Ans. 0.0202] Solution:  EMBED Equation.3 , ( = 19000, n = 30, ( = 2000, standard error of the mean ((x) =  EMBED Equation.3 =  EMBED Equation.3  consider the standard normal probability distribution, as follows: Z =  EMBED Equation.3  Now P(earnings will average more than Rs.19750 annually) = P( EMBED Equation.3  = P(Z > 2.05) = 1- P(Z ( 2.05) = 1- F(2.05) = 1 0.9798 = 0.0202 Therefore we have determined that there is slightly more than a 2% chance of average earnings more than Rs.19750 annually in a group of 30 letters. Hence the solution. Problem 27# If a gallon can of paint covers on the average 513.3 square feet(Ft2.) with a standard deviation(s.d.) of 31.5 square feet(Ft2.). what is the probability that the mean area covered by a sample of 40 of these 1 gallon cans will be anywhere from 510 to 520 square feet(Ft2.)? [Ans.0.6553] Solution: n = 40, ( = 513.3 and ( = 31.5 Let Z =  EMBED Equation.3  And Z =  EMBED Equation.3  P(510 < EMBED Equation.3 < 520) = P(-0.66 < Z < 1.34) = F(1.34)- F(-0.66) = F(1.34) 1 +F(0.66) = 0.9099 - 1 + 0.7454 = 0.6553 We obtain the probability 0.6553 note that if  EMBED Equation.3  turned out to be much less than 513.3, say less than 500 this might cause serious doubt whether the sample actually came from a population having ( = 513.3 and ( = 31.5. the probability of obtaining such a small value i.e., Z < -2.67 is only 0.0038. Hence the solution. Problem 28# A random sample of 100 is taken from an infinite population having the mean ( = 76 and the variance = (2 = 256. Find the probability that  EMBED Equation.3  will be between 75 and 78? [Ans. 0.6268] Solution: n = 100, ( = 76 and ( = 256 P(75 <  EMBED Equation.3  < 78) = P EMBED Equation.3 ( EMBED Equation.3  = P(-0.625 < Z < 1.25) = F(1.25) F(-0.625) = F(1.25) 1 + F(0.625) = 0.8944 1 + 0.7324 = 0.6268 Hence the solution. Problem 29# If two independent random samples of size n1 = 13 and n2 = 7 are taken from a normal population. What is the probability that the variance of the first sample will be atleast four times as that of the second sample? [Ans. 4.00] Solution: Given n1 = 13 and (1= n1 1 = 12 ; N2 = 7 and (2= n2 1 = 6 S12 = 4S22 Now F =  EMBED Equation.3 =  EMBED Equation.3  This value of F follows F-distribution with (1= n1 1 = 12 and (2= n2 1 = 6 degrees of freedom Hence from tables we get F0.05 (12,6) = 4.00 Hence the required probability is 0.05. Hence the solution. Problem 30# If two independent random samples of size n1 = 26 and n2 = 8 are taken from a normal population. What is the probability that the variance of the second sample will be atleast 2.4 times as that of the first sample? [Ans. 0.05] Solution: Given n1 = 20 and n2 = 8, (1= 19 and (2= 7 S12 = (2.4)S22 Now F ((19,7) = 2.4 ( ( = 0.05 Hence the solution. Problem 31# If the actual amount of instant coffee which a filing machine puts into 6-ounce jars is r. v. having a normal distribution with s.d. 0.05 ounce and if only 3% of the jars are to contain less than 6 ounces of coffee, what must be the mean fill of these jars? [Ans. (=6.094] Solution: Let X be the actual amount of coffee put into the jars, X ( N((, 0.05) Given P(X < 6) = 0.03 P(-( <  EMBED Equation.3 = 0.03 P(-( <  EMBED Equation.3  0.5- P(0 < Z <  EMBED Equation.3  P(0 < Z <  EMBED Equation.3  from table of areas P(0 < Z < 1.808) = 0.47 Implies  EMBED Equation.3 ( ( = 6.094 ounces. Hence the solution. Problem 32# A manufacturer of a certain type of synthetic fishing line has found from long experience of testing that the breaking strength of his product has an approximate normal distribution with a mean of 30 pounds( lbs. ) and a standard deviation of 4 pounds( lbs. ). A time and money saving change in the manufacture process of the product is tried. A sample of 25 testing length pieces of the new process line is taken and tested with a resulting sample mean of 28 pounds(lbs.) What is the probability of obtaining a mean as low as 28 if the process has had no harmful effect on breaking strength? [Ans. 0.006] Solution: Let X be the breaking strength of a randomly selected piece of line and if X ( N(30, 4) and n = 25, ( EMBED Equation.3 (or  EMBED Equation.3 )=30, ( EMBED Equation.3 (or s) =  EMBED Equation.3  Then P( EMBED Equation.3  ( 28) = P( EMBED Equation.3 = P(Z ( - 2.5) = F( - 2.5) = 1 F(2.5) = 1 0.9938 = 0.006 Thus there is a very small chance of obtaining a sample mean as low as 28 if ther had been no change in the quality of the line due to the new process.Hence the solution. Problem 33# An Urn contains 1000 white and 2000 black balls. If X denotes the number of white balls when 300 balls are drawn without replacement, then find P(180 < X < 120)? [Ans. 0.9858] Solution: clearly X ( B.Dn=(300, 1/3) If p = P(the ball drawn is white) = 1/3 Mean ( = np = 300 X 1/3 = 100 Variance = (2 = npq = 200 /3 Since n = 300 is large the required probability is P(80 < Z < 120) = P( EMBED Equation.3 = P(-2.45 < Z < 2.45) = 0.9858 Hence the solution. Problem 34# Two movie theatres compete for 900 visitors. Suppose each visitor chooses one of the two balls independent of the choice of the other visitors; how many seats should each theatre have so that the probability of turning away any visitor for lack of seats is less than 1%? [Ans. 489] Solution: clearly X ( B.Dn=(900, 1/2) If p = P(i=1 to 900) = 1/2 Mean ( = np = 900 X 1/2 = 450 Variance = (2 = npq = 900 /2 X 2 = 225 Since n = 900 is large the required probability is a N(0,1) random variable Now P(-2.58 (  EMBED Equation.3  P( EMBED Equation.3 = P(-2.45 < Z < 2.45) = 0.9858. So the required number of seats is 489. Hence the solution. Problem 35# Let X be a random variable where (x is unknown as (x2 = 0.25 i.e.,1/4 Find out how large a random sample must be taken in order that the probability will be at test 0.95 and the sample mean  EMBED Equation.3  will lies within 0.25 of the population mean? [Ans. 80] Solution: we have (x2 = 0.25, ( = 0.25 and 1 -  EMBED Equation.3  Therefore 0.05 >  EMBED Equation.3 and n >  EMBED Equation.3 =  EMBED Equation.3 Hence the solution. Problem 36# If a random sample of size n is selected from the finite population that consists of the integers 1,2,3,. . . ,N show that (i) the mean  EMBED Equation.3  is  EMBED Equation.3  (ii) the variance of  EMBED Equation.3  is  EMBED Equation.3  (iii) the mean and the variance of Y = n.  EMBED Equation.3  are E(Y) =  EMBED Equation.3  and the var(Y) =  EMBED Equation.3 ? Solution: (i)  EMBED Equation.3  (( =  EMBED Equation.3  (ii) Variance((2) =  EMBED Equation.3  =  EMBED Equation.3  ((2 =  EMBED Equation.3  (Var( EMBED Equation.3 ) =  EMBED Equation.3  (iii)(y =  EMBED Equation.3  Var(Y) =  EMBED Equation.3  ( Var(Y) =  EMBED Equation.3  Problem 37# How many different samples of size n =3 can be drawn from a finite population of size (a) N =12 (b) N = 20 (c) N = 50 [Ans. a) 220, b) 1140 c) 19600] Solution: a)12C3 =  EMBED Equation.3 ; b) 20C3 =  EMBED Equation.3 ; c) 50C3 =  EMBED Equation.3 ; Hence the solution. Problem 38# What is the probability of each possible sample if (i) a random sample of size n =4 is to be drawn from a finite population of size N = 12 (ii) a random sample of size n = 5 is to be drawn from a finite population of size N = 22? [Ans. a) 1/495 b) 1/77] Solution: (i)  EMBED Equation.3  (ii)  EMBED Equation.3  Hence the solution. Problem 39# Independent random samples of size n1 = 30 and n2 = 50 are taken from two normal populations having the means (1 = 78 and (2 = 78 and the variances (12 and (22. Find the probability that the mean of the first sample will exceed that of the second sample by at least 4.8? [Ans. 0.2743] Solution: clearly ( EMBED Equation.3 = 78 75 = 3 ( EMBED Equation.3 =  EMBED Equation.3  P( EMBED Equation.3 > 3) = P(Z >  EMBED Equation.3 = P(Z > 0.6) = 0.2743. Hence the solution. Problem 40# If S1 and S2 are the variances of independent random samples of size n1 = 61 and n2 = 31 from normal population with (12 = 12 and (22 = 18 Find  EMBED Equation.3  [Ans. 0.05] Solution: Let  EMBED Equation.3  Consider  EMBED Equation.3  = P(F > 1.74) for 60 + 30 d.o.f. = 0.05 Hence the solution. Chapter 2 Objective bits III Sampling Distributions by N. V. Nagendram 01. A sample consists of ___________________________ [ Ans. any part of population] 02. Another name of population is ___________________ [Ans. Universe] 03. The number of possible samples of size n out of N population units without replacement is ___________________ [Ans. NCn] 04. The number of possible samples of size n from a population of N units with replacement is ___________________ [Ans.  EMBED Equation.3 ] 05. Probability of anyone sample of size n being drawn out of N units is ___________________ [Ans.  EMBED Equation.3 ] 06. Probability of including a specified unit/ item in a sample of size n selected out of N units is___________________ [Ans.  EMBED Equation.3 ] 07. Having sample observations x1, x2, x3, . . ., xn the formula for variance is ___________________ [Ans. s2 =  EMBED Equation.3 ] 08. Sample mean formula ___________________ [Ans.  EMBED Equation.3 =  EMBED Equation.3 ] 09.  EMBED Equation.3  is called ___________________ [Ans. Finite population correction factor] 10. The discrepencies between sample estimate and population parameter is the ___________________ [Ans. Sampling Error] 11. If the observations recorded on five sampled items are 3,4,5,6,7 the sample variance is ___________________ [Ans. 2.5] 12. A population consisting of all real numbers is an example of [Ans. An infinite population] 13. Standard deviation of all possible estimate from samples of fixed size is called ___________________ [Ans. Standard error] 14. A population parameter is a ___________________ associated with the entire population [Ans. descriptive or statistical] 15. If  EMBED Equation.3  is the mean of a random sample size n taken from a population nearly normal having mean ( and the finite variance (2 then Z =  EMBED Equation.3  Is a random variable following as n tends to infinite i.e. n (( [Ans. standard normal distribution] 16. Standard error of the statistic sample mean  EMBED Equation.3 ___________________ [Ans.  EMBED Equation.3 ] 17. If x1, x2, x3, . . ., xn constitute a random sample from an infinite population with the mean ( and the variance (2 then (( EMBED Equation.3 ) = ____________ and (2( EMBED Equation.3 )= _____________[Ans. (,  EMBED Equation.3 ] 18. If  EMBED Equation.3 is the mean of a random sample from a finite population size N with the mean ( and the variance (2 then (( EMBED Equation.3 ) = ____________ and (2( EMBED Equation.3 )= ______ [Ans. (,  EMBED Equation.3 ] 19. t1-( = __________________ [Ans. - t(] 20. F1-(((1, (2) = ________________ [Ans.  EMBED Equation.3 ] Chapter 1 PROBABILITY DISTRIBUTION Tutorial 16 Probability Density Function Problems REVISION by: N.V.Nagendram Problem #1 If E(X) = 1, E(X2) = 4, find the mean and variance of Y = 2x -3? [Ans. Var = 12] Problem #2 A continuous random variable X has the p.d.f. given by f(x) = kx2, 0 ( x ( 1. Find the value of k. with this value of k find P( x <  EMBED Equation.3 ) and P( x (  EMBED Equation.3 )? [Ans.  EMBED Equation.3 , EMBED Equation.3 ] Problem #3 The probability density p(x) of a continuous random variable is given by p(x) = y0 e-| x | , - ( < x < (, prove that y0 =  EMBED Equation.3  find the mean and variance of the distribution? [Ans. var = 2] Problem #4 A continuous random variable X has the p.d.f. given by f(x) = kx, 0 ( x ( 1 = k, 1 ( x ( 2 = -x+3k, 2 ( x ( 3 = 0 otherwise. Find the value of k. Also calculate P(X ( 1.5)? [Ans.  EMBED Equation.3 ] Problem #5 Given that f(x) =  EMBED Equation.3 is a probability distribution function for a random variable X, that can take on the values x = 0,1,2,3 and 4 (i) find k (ii) mean and variance of x? [Ans. 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hocCJOJQJUVaJjhocCJOJQJUaJUhoc5CJOJQJaJhmhoc5CJOJQJaJhocCJH*OJQJaJ jshocCJOJQJaJhocCJOJQJaJ jmhocCJOJQJaJ (a) is the function f(x), defined as follows, a density function? f(x) = 0 x < 2 =  EMBED Equation.3 (3 + 2x) -2 ( x ( 4 = 0, x > 4 (b) Find the probability that a variate having this density will fall in the interval 2 ( x ( 3? [Ans. a) 1b)  EMBED Equation.3 ] Problem #7 Find the constant k so that function F(x) is defined as follows may be a density function: f(x) =  EMBED Equation.3  a ( x ( b = 0 elsewhere. Find also the cumulative distribution function of the random variable X and K satisfies the requirements for f(x) to be a density function? [Ans. k = b-a, F(x) = 1] Chapter 1 PROBABILITY DISTRIBUTION Tutorial 17 Probability Density Function Problems REVISION by: N.V.Nagendram Problem #8 If X is a continuous random variable with p. d. f. given by F(x) = kx 0 ( x ( 2 = 2k 2 ( x ( 4 = -kx + 6k 4 ( x ( 6 Find the value of k and mean value of X. [Ans. k= EMBED Equation.3 , (=E(X) = 3] Problem #9 (a) verify that the following is a distribution function: F(x) = 0 x < - a =  EMBED Equation.3 ( EMBED Equation.3 +1) -a ( x ( a = 1 x > 1 (b) show that F(x) = 0 - ( < x < 0 = 1- e x 0 ( x < ( is possible distribution function and find the density function? [Ans. a)1 b) 1] Problem #10 A random process gives measurements X between 0 and 1 with a probability density function f(x) = 12 x3 21 x2 + 10 x, 0 ( x ( 1 = 0 otherwise. (i) find P(X (  EMBED Equation.3 ) and P(X >  EMBED Equation.3 ) (ii) Find a number k such that P( X ( k) =  EMBED Equation.3 ? [Ans. a) EMBED Equation.3 ,b) EMBED Equation.3 , EMBED Equation.3 k = 0.45] Problem #11 The probability distribution function of a random variable X is f(x) = x 0 ( x ( 1 = 2 x 1 ( x ( 2 = 0 x ( 2 compute the cumulative distribution function of X? [Ans. F(x) = 1] Problem #12 The frequency function of a continuous random variable is given by f(x) = y0 x (2 x), 0 ( x ( 2. Find the value of y0, mean and variance of X ? 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IdZQ~X 'យ ŵUSoDd Xb P c $AP? ?3"`?O2\>+RPW|8^%`!0>+ObjInfouEquation Native _1444740084xFݡLQݡLQOle w61{0 f X/Y (x/y)"x"R FMicrosoft Equation 3.0 DS Equation Equation.39qh2Tq0 F X/Y (CompObjwyfObjInfozEquation Native _1444804720}FݡLQݡLQx/y) = f X/Y (t/y)dt"x"R -"" +" FMicrosoft Equation 3.0 DS Equation Equation.39q>p6 t Ole CompObj|~fObjInfoEquation Native #_1444804730lFݡLQݡLQOle CompObjfObjInfo   !$'(),/012347:;<=>?@CFGHKNORUVY\]`cfijklmnopqty~ FMicrosoft Equation 3.0 DS Equation Equation.39q>exu!v E(X)=xP(X=x) x "Equation Native _1444804807FݡLQݡLQOle CompObjf FMicrosoft Equation 3.0 DS Equation Equation.39q>H6 u E(X k )=x k P(X=x) x " FMicrosoft Equation 3.0 DS EqObjInfo Equation Native  _1444804962FݡLQݡLQOle CompObjfObjInfoEquation Native _1444805044FݡLQݡLQuation Equation.39q>v86(t' E(X)=xf(x)dx """ +" FMicrosoft Equation 3.0 DS Equation Equation.39qOle CompObjfObjInfoEquation Native >’)Dq' E(X k )=x k f(x)dx """ +" FMicrosoft Equation 3.0 DS Equation Equation.39q_1444805702FݡLQݡLQOle CompObjfObjInfoEquation Native  V_1444806429FݡLQݡLQOle "CompObj#f>:863t2 = V(X)  FMicrosoft Equation 3.0 DS Equation Equation.39q>863t2 E(x+y)=(x i +ObjInfo%Equation Native &_1444806869FݡLQݡLQOle *y j ) i=1m " j=1n " P ij FMicrosoft Equation 3.0 DS Equation Equation.39q>›a4v2 =CompObj+fObjInfo-Equation Native ._1444806894FݡLQݡLQx i  i=1m " j=1n " P ij +y j  i=1m " j=1n " P ij  FMicrosoft Equation 3.0 DS Equation Equation.39qOle 5CompObj6fObjInfo8Equation Native 9>63{2 =x i  i=1m "  j=1n " P ij ()+y j  i=1m "  j=1n " P ij () FMicrosoft Equation 3.0 DS Equation Equation.39q>`c4<|2 =x ii " P ij +y jj " _1444806982FݡLQݡLQOle ACompObjBfObjInfoDEquation Native E_1444807237FݡLQ'LQOle ICompObjJfP ij  FMicrosoft Equation 3.0 DS Equation Equation.39q>s86%t$ P ijj=1ton " =P jObjInfoLEquation Native M_1444807273F'LQ'LQOle P FMicrosoft Equation 3.0 DS Equation Equation.39q>v &Dq$ P iji=1tom " =PiCompObjQfObjInfoSEquation Native T_1444808010F'LQ'LQOle WCompObjXfObjInfoZEquation Native [ FMicrosoft Equation 3.0 DS Equation Equation.39q>86't& E(xy)=x i y j P iji " j " FMicrosoft Equation 3.0 DS Eq_1444808265F'LQ'LQOle ^CompObj_fObjInfoauation Equation.39q>b86/t. P ij =P i P j FMicrosoft Equation 3.0 DS Equation Equation.39qEquation Native b~_1444808794F'LQ'LQOle dCompObjefObjInfogEquation Native hx_1444893101F'LQ'LQOle r>\|0Dq. x i P i y j P j = i " j "  i " x i P j  j " y j P j =P ii " x i E(y)=E(y) i " P i x i =E(x)E(y) FMicrosoft Equation 3.0 DS Equation Equation.39q,8 14CompObjsfObjInfouEquation Native v6_1444893143F'LQ'LQOle wCompObjxfObjInfozEquation Native {6 FMicrosoft Equation 3.0 DS Equation Equation.39q,h;g 24 FMicrosoft Equation 3.0 DS Equation Equation.39q_1444893218F'LQ'LQOle |CompObj}fObjInfoEquation Native 6_1444893544F'LQ'LQOle CompObjf,%;$ 24 FMicrosoft Equation 3.0 DS Equation Equation.39q,x4;3 24 FMicrosoft Equation 3.0 DS EqObjInfoEquation Native 6_1444893586F'LQ'LQOle CompObjfObjInfoEquation Native 6_1444893596F'LQ'LQuation Equation.39q,W6T>3 24 FMicrosoft Equation 3.0 DS Equation Equation.39q,Y6C3 12Ole CompObjfObjInfoEquation Native 6_1444902277F'LQ'LQOle CompObjfObjInfo FMicrosoft Equation 3.0 DS Equation Equation.39q,#;" Mean==x.b(x;n,p) x=0n "Equation Native _1444902499F'LQ'LQOle CompObjf FMicrosoft Equation 3.0 DS Equation Equation.39q,C#>" =x.n!x!(n"x)!p xObjInfoEquation Native __1444902575F'LQ'LQOle  q x=0n "  (n"x) FMicrosoft Equation 3.0 DS Equation Equation.39q,G!;  CompObjfObjInfoEquation Native c_1444903210F'LQ'LQ=n!(x"1)!(n"x)!p x q x=1n "  (n"x) FMicrosoft Equation 3.0 DS Equation Equation.39qOle CompObjfObjInfoEquation Native ,oH&;% =np(n"1)!(x"1)!(n"x)!p x"1 q x=1n "  (n"x)_1444903372F'LQ'LQOle CompObjfObjInfo FMicrosoft Equation 3.0 DS Equation Equation.39q,jt(T>% =np, (n"1) C y p y q y=0n"Equation Native _1444903399F'LQ'LQOle CompObjf1 "  (n"y"1) FMicrosoft Equation 3.0 DS Equation Equation.39q, &F'LQ'LQOle =np(q+p) (n"1) FMicrosoft Equation 3.0 DS Equation Equation.39q®3 E(X) 2 =x 2 b(xCompObjfObjInfoEquation Native _1444925038 F'LQ'LQ,n,p) x=0n " FMicrosoft Equation 3.0 DS Equation Equation.39qT6 =Ole CompObj fObjInfo Equation Native [x(x"1)+x]b(x,n,p) x=0n " FMicrosoft Equation 3.0 DS Equation Equation.39qTh: _1444925147F'LQ'LQOle CompObj fObjInfoEquation Native p_1444925301F'LQ'LQOle CompObjf=x(x"1)b(x,n,p)+ x=0n " xb(x,n,p) x=0n " FMicrosoft Equation 3.0 DS Equation Equation.39qObjInfoEquation Native g_1444925453  F'LQ'LQOle K!;  =x(x"1)n!x!(n"x)!p x q (n"x) + x=0n "CompObjfObjInfoEquation Native c_1444925485F'LQ'LQ FMicrosoft Equation 3.0 DS Equation Equation.39qGx!T>  =n!(x"2)!(n"x)!p x q (n"x) +np x=2n " FMicrosoft Equation 3.0 DS Equation Equation.39q±!|=  =n(n"1)p 2 f     ST !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRUVWYXZ\[]^_a`bcdehgijlknmpoqrtsvuxwyz{|}~Ole CompObjfObjInfoEquation Native     !"#$'*+.12589:;>CFGHILOPSVY^abehijklorstux{|}~(n"2)!(x"2)!(n"x)!p (x"2) q (n"x) +np x=2n " FMicrosoft Equation 3.0 DS Equation Equation.39q_1444925651%"F'LQ'LQOle CompObj!#fObjInfo$­P#;" =n(n"1)p 2 (n"2)!y!(n"2"y)!p y q (n"2"y) +np y=0n"2Equation Native  _1444925753'F'LQ'LQOle CompObj&(f " FMicrosoft Equation 3.0 DS Equation Equation.39q€ #T>" =n(n"1)p 2 ObjInfo)Equation Native _1444925910M,F'LQ'LQOle , (n"2) C y p y q (n"2"y) +np y=0n"2 " FMicrosoft Equation 3.0 DS Equation Equation.39qCompObj+-fObjInfo.Equation Native  _14449262001F'LQ'LQ+;* =n(n"1)p 2 (q+p) n"2 +np FMicrosoft Equation 3.0 DS Equation Equation.39qOle %CompObj02&fObjInfo3(Equation Native )•k;j E(X 2 )=n(n"1)p 2 +np FMicrosoft Equation 3.0 DS Equation Equation.39q_1444926240/C6F'LQ'LQOle ,CompObj57-fObjInfo8/Equation Native 0_1444926348;F'LQ'LQOle 3CompObj:<4f†hkT>j  2 =E(X 2 )"[E(X)] 2 FMicrosoft Equation 3.0 DS Equation Equation.39qpkBj ObjInfo=6Equation Native 7_1444927442*u@F'LQ'LQOle <=n(n"1)p 2 +np"(np) 2 FMicrosoft Equation 3.0 DS Equation Equation.39qRPW|JX+xK#ALc6<.FEP[,XD =QV8RHDDÿ@l{3t`|o08n/,w1ox֯,ŕ_C֕/-I6tNsN?$`j^s6#j?p(9s+iLEkqw)Lj%]z7e7 v|$6i[ZpU0 >W\HeXt[xsZmj ey3դ'T\rw^SuRz'M<GQŝ`!eOxK}IhWnVH1;hʼyQy-JyOm\<)y}NwY3epN9=zn}7NaҴˤ3Uxˌެi-1{2SbJ3D3 s6zDd \hb Q c $AQ? ?3"`?P2S"v d3.Dm _Kp%`!S"v d3.Dm _K` @0|xR=KA}0|R)BbB! 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