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Unformatted text preview: 1 STAT 1301 Probability & Statistics Lecture 11 Lecture 11 Negative Binomial, Geometric, and Hypergeometric Distributions Introduction • The geometric , negative binomial and hypergeometric distributions are all closely related to the binomial distribution. The geometric and negative binomia distributions • The geometric and negative binomial distributions arise from fixing the number of S ’s and letting the number of trials to be random. • The hypergeometric distribution is the exact probability model for sampling without replacement from a finite dichotomous ( S , F ) population. Negative Binomial Dist’n • The experiment consists of a sequence of independent trials. • Each trial results is either S or F . • The probability of success, p , is constant from trial to trial. • Trials are performed until a total of r successes have been observed, where r is a pre-specified positive integer. Negative Binomial RV • X , the total number of trials required, is called a negative binomial rv . • Possible values of X are r , r +1, r +2, … 2 pmf • Denote by nb ( x; r, p ) the pmf of X . Then ,... 1 , , ) 1 ( 1 ) , ; ( r r x p p x p r x nb r x r • Why? • Total # of trials = x ; The last trial must be a success. Among the first ( x-1) trials, there are ( r- 1) successes & x-r failures. 1 r Mean & Variance • If X has pmf nb ( x; r, p ), then ) 1 ( p r X V r X E . ) ( , ) ( 2 p X V p X E Example • A door-to-door salesperson is required to document 5 in-home visits each day. Suppose she has a 30% chance of being invited into any given home If she selects invited into any given home. If she selects, in advance, 10 addresses to visit. • What’s the probability her 5th success occurs on the 10th trial? • What’s the chance she requires fewer than 8 addresses to meet her quota?...
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