ࡱ> ')&M Lbjbj== :WWLl , D4n 4444444$I5 i78484oM4ooov4o4oon1.3b „ O2 .3c4042he8-Be8.3oSTAT 211 Handout 5 (Chapter 5) Joint Probability Distributions and Random Samples By using the following examples, the joint probability mass function for two discrete random variables and their properties, their marginal probability mass functions, the case for independent and dependent variables, their conditional distributions, expected value, variance, covariance, and correlation will be demonstrated. Properties will be explained. Quality audit records are kept on numbers of major and minor failures of circuit packs during-in of large electronic switching devices. They indicate that for a device of this type, the random variables X (the number of major failures) and Y (the number of minor failures) can be described at least approximately by the accompanying joint distribution. x y01200.150.050.0110.100.100.0420.100.140.0430.100.130.04 Find the marginal probability functions for X and Y. Are X and Y independent? Find the expected value and the variance of X. Find the expected value and the variance of Y. Find the Cov(X,Y) and Corr(X,Y). Find the conditional distribution for Y given that X=0 that is there are no circuit pack failures. What is the mean for the conditional distribution of Y given that X=0? Suppose that demerits are assigned to devices of this type according to the formula D=5X+Y. Find the expected value of D. Suppose that demerits are assigned to devices of this type according to the formula D=5X+Y. Find the variance of D. Suppose that demerits are assigned to devices of this type according to the formula U=Min(X,Y). Find the mean value and the variance of U. By using the following example, the joint probability density function for two continuous random variables and their properties, their marginal probability density functions, the case for independent and dependent variables, their conditional distributions, expected value, variance, covariance, and correlation will be demonstrated. Properties will be explained. Suppose that a pair of random variables, X and Y have the same joint probability density  EMBED Equation.3  Find the marginal probability density functions for X and Y. Are X and Y independent? Evaluate P(X+2Y(1). Find the expected value and variance of X. Find the expected value and variance of Y. Find the conditional probability function of x given y=0.6. Find the conditional probability function of y given x=0.4. What is E(X|Y=0.6)? Find the Cox(X,Y) and Corr(X,Y). Random Sample: The random variables X1, X2, .,Xn are said to form a random sample of size n if (i) The Xi's are independent random variables. (ii) Every Xi's has the same probability distribution. The sampling distribution of  EMBED Equation.3 : Discussion theoretically Exercise 5.42 (textbook): A company maintains 3 offices in a certain region, each staffed by two employees. Information concerning yearly salaries (1000s of dollars) is as follows: Office 1 1 2 2 3 3 Employee 1 2 3 4 5 6 Salary 19.7 23.6 20.2 23.6 15.8 19.7 Suppose two of these employees are randomly selected from among the six (without replacement). Determine the sampling distribution of the sample mean salary  EMBED Equation.3 . S={(1,2),(1,3),(1,4),(1,5),(1,6),(2,1),(2,3),(2,4),(2,5),(2,6),(3,1),(3,2),(3,4),(3,5),(3,6), (4,1),(4,2),(4,3),(4,5),(4,6),(5,1),(5,2),(5,3),(5,4),(5,6),(6,1),(6,2),(6,3),(6,4),(6,5)}. There are 30 outcomes.  EMBED Equation.3  when (1,2),(2,1),(1,4),(4,1),(2,6),(6,2),(4,6),(6,4) with 8 outcomes when (1,3),(3,1),(3,6),(6,3) with 4 outcomes when (1,5),(5,1),(5,6),(6,5) with 4 outcomes 19.70 when (1,6),(6,1),(4,5),(5,4),(2,5),(5,2) with 6 outcomes when (2,3),(3,2),(3,4),(4,3) with 4 outcomes when (2,4),(4,2) with 2 outcomes when (3,5),(5,3) with 2 outcomes  EMBED Equation.3  : 17.75 18 19.7 19.95 21.65 21.90 23.60 otherwise  EMBED Equation.3  : 4/30 2/30 6/30 4/30 8/30 4/30 2/30 0 E( EMBED Equation.3 )= EMBED Equation.3 =613/30=20.43 Suppose one of the three offices is randomly selected. Determine the sampling distribution of the sample mean salary  EMBED Equation.3  . S={(1,2),(2,1),(3,4),(4,3),(5,6),(6,5)}. There are 6 outcomes  EMBED Equation.3  21.65 when (1,2),(2,1) with probability 1/3 21.90 when (3,4),(4,3) with probability 1/3 17.75 when (5,6),(6,5) with probability 1/3 E( EMBED Equation.3 )=61.3/3=20.43 Population mean = (19.7+23.6+20.2+23.6+15.8+19.7)/6=20.43 Additional: The sampling distribution of the range of salaries for part (b) in the previous question is range: 3.4 3.9 p(range) : 2/6 4/6 E(range)=  EMBED Equation.3 =3.7333 where Population range is 23.6-15.8=7.8 The distribution of a linear combination of variables: Discussion theoretically Exercise 5.60: First two cars are economy brand and distributed N(20,4) Last three cars are name brand and distributed N(21,3.5) All five are independent. Y is a measure of the difference in efficiency between economy gas and name brand gas. E(Y)=  EMBED Equation.3 = EMBED Equation.3 =20-21=-1 Var(Y)=  EMBED Equation.3 = EMBED Equation.3 =2+1.6667=3.6667 P(Y(0)=P(Z(0.52)=0.3015 P(-1(Y(1)=P(Y(1)-P(Y<-1)=P(Z(1.05)-P(Z<0)=0.8531-0.5=0.3531 Exercise 5.66 : (a) 50, 10.308 (b) 0.0075 (c) 50 (d) 111.5625 (e) 50, 131.25 Exercise 5.68: Suppose the speed for the 1st airplane is X1 and the speed for the 2nd airplane is X2. (a) P(X1-X2 <5)=0.9616 (b) P(0(X2-X1(10)=0.0623 Exercise 5-69: (a) 2400 (b) with independence,1205 (c) 2400, 41.77 Central Limit Theorem: Let X1, X2,.,Xn be a random sample from a distribution with mean ( and variance (2. Then if n is sufficiently large (n>30),  EMBED Equation.3  has approximately a normal distribution with mean  EMBED Equation.3 and variance  EMBED Equation.3 and  EMBED Equation.3 has approximately a normal distribution with mean  EMBED Equation.3 and variance  EMBED Equation.3 . The larger the value of n, the better the approximation. 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