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sta4321-11 - Geometric and Negative Binomial Distributions...

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Unformatted text preview: Geometric and Negative Binomial Distributions Geometric Distribution Negative Binomial Distribution Geometric Distribution – Number of Failures to First Success When flipping a coin, we count the number of tails before the first heads appears. When setting off fireworks, we count the number of successfully fired fireworks before the first dud appears. When rolling two dice, we count the number of rolls before the dice sum to seven. Number of Failures to First Success (or equivalently, number of success to first failure) Arthur Berg Geometric and Negative Binomial Distributions 2/ 11 Geometric Distribution Negative Binomial Distribution Geometric Distribution – Number of Failures to First Success When flipping a coin, we count the number of tails before the first heads appears. When setting off fireworks, we count the number of successfully fired fireworks before the first dud appears. When rolling two dice, we count the number of rolls before the dice sum to seven. Number of Failures to First Success (or equivalently, number of success to first failure) Arthur Berg Geometric and Negative Binomial Distributions 2/ 11 Geometric Distribution Negative Binomial Distribution Geometric Distribution – Number of Failures to First Success When flipping a coin, we count the number of tails before the first heads appears. When setting off fireworks, we count the number of successfully fired fireworks before the first dud appears. When rolling two dice, we count the number of rolls before the dice sum to seven. Number of Failures to First Success (or equivalently, number of success to first failure) Arthur Berg Geometric and Negative Binomial Distributions 2/ 11 Geometric Distribution Negative Binomial Distribution Geometric Distribution built from Bernoulli rv’s Let Y 1 , Y 2 , . . . be iid Bernoulli( p ) random variables, i.e. Y i = 1 , w.p. p , w.p. 1- p . Let X be the random variable that counts the number of failures of the Bernoulli trials before the first success. Therefore if Y 1 = 1 then X = if Y 1 = 0 and Y 2 = 1 then X = 1 if Y 1 = 0, Y 2 = 0, and Y 3 = 1 then X = 2 . . . . . . Arthur Berg Geometric and Negative Binomial Distributions 3/ 11 Geometric Distribution Negative Binomial Distribution Geometric Distribution built from Bernoulli rv’s Let Y 1 , Y 2 , . . . be iid Bernoulli( p ) random variables, i.e. Y i = 1 , w.p. p , w.p. 1- p . Let X be the random variable that counts the number of failures of the Bernoulli trials before the first success. Therefore if Y 1 = 1 then X = if Y 1 = 0 and Y 2 = 1 then X = 1 if Y 1 = 0, Y 2 = 0, and Y 3 = 1 then X = 2 . . . . . . Arthur Berg Geometric and Negative Binomial Distributions 3/ 11 Geometric Distribution Negative Binomial Distribution Geometric Distribution built from Bernoulli rv’s Let Y 1 , Y 2 , . . . be iid Bernoulli( p ) random variables, i.e....
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sta4321-11 - Geometric and Negative Binomial Distributions...

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