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# Lecture6standard - Statistics 511: Statistical Methods Dr....

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Statistics 511: Statistical Methods Dr. Levine Purdue University Fall 2011 Lecture 6: The Binomial, Hypergeometric, Negative Binomial and Poisson Distributions Devore: Section 3.4-3.6 Sept, 2011 Page 1

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Statistics 511: Statistical Methods Dr. Levine Purdue University Fall 2011 Binomial Experiment 1. The experiment consists of a sequence of n trials, where n is ﬁxed in advance of the experiment. 2. The trials are identical, and each trial can result in one of the same two possible outcomes, which are denoted by success (S) or failure (F). 3. The trials are independent 4. The probability of success is constant from trial to trial and is denoted by p. Given a binomial experiment consisting of n trials, the binomial random variable X associated with this experiment is deﬁned as X = the number of Ss among n trials Sept, 2011 Page 2
Statistics 511: Statistical Methods Dr. Levine Purdue University Fall 2011 Example where the experiment is not binomial I Consider 50 restaurants to be inspected; 15 of them currently have at least one serious health code violation while the rest have none. There are 5 inspectors, each of whom will inspect 1 restaurant during the coming week. The restaurant names are sampled as slips of paper without replacement ; i th trial is a success if the restaurant has no violations where = 1 ,..., 5 . Then P ( S on the 1st ) = 35 50 = . 70 Sept, 2011 Page 3

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Statistics 511: Statistical Methods Dr. Levine Purdue University Fall 2011 Example where the experiment is not binomial II Similarly, P ( S on the 2nd ) = P ( SS ) + P ( FS ) = . 70 However, P ( S on the 5h trial | SSSS ) = 31 46 = . 67 while P ( S on the 5h trial | FFFF ) = 35 46 = . 76 If the sample size n is at most 5% of the population size, the experiment can be analyzed as though it were exactly a binomial experiment. Sept, 2011 Page 4
Statistics 511: Statistical Methods Dr. Levine Purdue University Fall 2011 Binomial pmf Because the pmf of a binomial rv X depends on the two parameters n and p, we denote the pmf by b(x;n,p). The binomial pmf is b ( x ; n,p ) = ( n x ) p x (1 - p ) n - x x = 0 , 1 , 2 ,...,n 0 otherwise Sept, 2011 Page 5

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Statistics 511: Statistical Methods Dr. Levine Purdue University Fall 2011 Expected value and variance of a binomial RV Let X 1 ,...,X n be mutually independent Bernoulli random variables, each with success probability p. Then, Y = n i =1 X i is a binomial random variable with pmf b ( x ; n,p ) . The expected value is E Y = E n X i =1 X i ! = n X i =1 E X i = np The variance is V ( Y ) = V n X i =1 X i ! = n X i =1 V ( X i ) = n X i =1 p (1 - p ) = np (1 - p ) Sept, 2011 Page 6
Dr. Levine Purdue University Fall 2011 Example A card is drawn from a standard 52-card deck. If drawing a club is considered a success, ﬁnd the probability of 1. exactly one success in 4 draws (with replacement) 2. no successes in 5 draws (with replacement). Sept, 2011

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## This note was uploaded on 02/20/2012 for the course STAT 511 taught by Professor Bud during the Fall '08 term at Purdue University-West Lafayette.

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Lecture6standard - Statistics 511: Statistical Methods Dr....

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