ࡱ> z@ nbjbj.. \DDD(4llll```8|,hƂ܂܂$Rkj*AF?@FF*ll܂;k888Fpll܂(8F88vl,( G`X|,0((llll(vT8ID**$L # Binomial MTB > #P = 0.05 and N = 16 MTB > CDF; SUBC> Binomial 16 .05. Cumulative Distribution Function Binomial with n = 16 and p = 0.0500000 x P( X <= x ) 0 0.4401 1 0.8108 2 0.9571 3 0.9930 4 0.9991 5 0.9999 6 1.0000 MTB > # Inverse Binomial MTB > MTB > INVCDF .1247; SUBC> Binomial 16 .05. Inverse Cumulative Distribution Function Binomial with n = 16 and p = 0.0500000 x P( X <= x ) x P( X <= x ) 0 0.0000 0 0.4401 Poisson MTB > # Poisson with mean of 5 MTB > MTB > CDF; SUBC> Poisson 5. Cumulative Distribution Function Poisson with mu = 5.00000 x P( X <= x ) 0 0.0067 1 0.0404 2 0.1247 3 0.2650 4 0.4405 5 0.6160 6 0.7622 7 0.8666 8 0.9319 9 0.9682 10 0.9863 11 0.9945 12 0.9980 13 0.9993 14 0.9998 15 0.9999 16 1.0000 CONTINUOUS RANDOM VARIABLES A continuous random variable can assume values represented by a continuous interval of numbers Examples current in a copper wire (variation from current source, temperature change) diameter of a bolt (variation from calibration, tool wear, raw materials) time to complete a machining operation length of time to play a set of badminton heights, weights, lengths, etc, Note the probability of selecting exact values cannot be measured; instead we are concerned with the probability of an interval of values, and tabular forms are no longer possible. Instead we use a function, referred to as the probability density function. Probability Density Function The function f(x) is a probability density function for the continuous random variable X, defined over the real numbers R if: f(x) > 0 for all x that are elements of R ( ( f(x)dx = 1 = ( b P(a < x < b) > ( f(x)dx a Note that consequence of X being a continuous random variable is that P(X =x) = 0 and when evaluating the probability of an interval, it is not necessary to consider the equality sign i.e. P(a < X < b) = P(a < X < b). Example The lead concentration in gasoline ranges from 0.2 to 0.6 grams per liter. Define the random variable X. X: grams of lead per liters The density of the random variable is given by f(x) = kx - 1 for 0.2 < x < 0.6 f(x) = 0 otherwise Find the value of k. What is the probability that a liter of gas will have between .3 and .5 grams of lead? [Ref. Milton and Arnold] Solution According to the definition ( ((kx-1)dx = 1 ( .2 . 6 ( ( 0dx + ((kx-1)dx + ( 0dx = 1 -( ..2 .6 Cumulative Distribution Function An alternate way of describe a continuous random variable is the cumulative distribution function (cdf). Assume X is a random variable. The function F(x) = P[X < x] x ( f(x) dx for -( < x < ( -( Example B For the distribution function of Example a, find the cumulative density function. Mean or Expected Value of a Continuous RV ( E(X) = (x f(x)dx -( Variance/Standard Deviation of a Continuous RV ( Var (X) = ((x - E(X))2 f(x)dx -( NORMAL DISTRIBUTIONS Family of distributions, all with the same general shape. Symmetric about the mean The y-coordinate (height) specified in terms of the mean and the standard deviation of the distribution Normal Probability Density For all x Note: e is the mathematical constant, 2.718282 ... Standard Normal Distribution The normal distribution with m =0 and s =1 is called the standard normal. For all x:  Transformations Normal distributions can be transformed to the standard normal. We use what is called the z-score, which is a value that gives the number of standard deviations that X is from the mean. Standard Normal Table Use the table in the text to verify the following. 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What proportion of the jar contain: a) Less than 6.06 oz? b) More than 6.09 oz? c) Less than 6 oz? Normal Example - part a) Assume m = 6 and s = 0.03. The problem requires us to find P(X < 6.06) Convert x = 6.06 to a z-score z = (6.06 - 6)/.03 = 2 and find: P(z < 2) = .9773 So 97.73% of the jars have less than 6.06 oz. Normal Example - part b) Again m = 6 and s = 0.03. The problem requires us to find: P(X > 6.09) Convert x = 6.09 to a z-score z = (6.09 - 6)/.03 = 3 and find P(z > 3) = 1- P(x < 3) = 1- .9987= 0.0013 So 0.13% of the jars have more than 6.09oz. Normal Distribution/ MTB MTB > # Use Calc\Probability Distributions MTB > # Normal Distribution MTB > CDF -1.77; SUBC> Normal 0.0 1.0. Cumulative Distribution Function Normal with mean = 0 and standard deviation = 1.00000 x P( X <= x ) -1.7700 0.0384 MTB > PDF -1.77; SUBC> Normal 0.0 1.0. Probability Density Function Normal with mean = 0 and standard deviation = 1.00000 x f( x ) -1.7700 0.0833 MTB > InvCDF .0833; SUBC> Normal 0.0 1.0. Inverse Cumulative Distribution Function Normal with mean = 0 and standard deviation = 1.00000 P( X <= x ) x 0.0833 -1.3832 MTB > CENTRAL LIMIT THEOREM specifies a theoretical distribution formulated by the selection of all possible random samples of a fixed size n a sample mean is calculated for each sample Sampling Distribution Of The Mean The mean of the sample means is equal to the mean of the population from which the samples were drawn. The variance of the distribution is s divided by the square root of n. (the standard error.) Standard Error Standard Deviation for the Distribution of Sample Means  EMBED Equation.3  Central Limit Theorem Consider a population with mean m and standard deviation s. Draw a random sample of n observations from this population where n is a large number (n> 30). Find the mean EMBED Equation.3 for each and every sample. The distribution of the sample means EMBED Equation.3 will be approximately normal. This distribution is called the Sampling Distribution of the Means or the Distribution of Sample Means. The mean and standard deviation (called the standard error) of the Distribution of Sample Means is: EMBED Equation.3The mean of the Sampling Distribution equals the mean of the PopulationEMBED Equation.3The standard error equals the standard deviation of the population divided by the square root of the sample size. 6. The approximation becomes more accurate as n becomes large. Example of CLT A certain brand of tires has a mean life of 25,000 miles with a standard deviation of 1600 miles. What is the probability that the mean life of 64 tires is less than 24,600 miles? 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