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Unformatted text preview: Chapter 5 Special Discrete and Continuous Distributions 5.1 The Bernoulli and Binomial distributions Definition 5.1 An experiment is called a Bernoulli trial if the outcome can be classified as either a “success” or “failure”. In the Bernoulli trial, let p = P (success), and X = 1 if the outcome is a success if the outcome is a failure Then x 1 f X ( x ) p q (1) where q = 1 p . • If X is the number of successes that occur in n independent Bernoulli trials, then X is called a binomial random variable with parameters ( n,p ), denoted by X ∼ B ( n,p ). X is also said to have binomial distribution with parameters n and p . • Suppose X ∼ B ( n,p ). Then, the probability density function of X is b ( x ; n,p ) = n x p x (1 p ) n x , x = 0 , 1 ,...,n • If X ∼ B ( n,p ), then EX = np, Var( X ) = npq, where q = 1 p 51 Example 5.1 A man claims to have extrasensory perception (ESP). As a test, a fair coin is flipped 109 times, and he is asked to predict the outcome in advance. The man gets 7 out of 10 correct. What is the probability that he would have done at least this well if he had no ESP? Solution : Example 5.2 A multiplechoice test consists of 20 questions each with 4 possible answers of which only 1 is correct. Suppose that a student knows the correct answers for 70% of questions, and the student randomly chooses an answer if he doesn’t know the correct answer. (i) what is the probability that the student will get at least 16 correct answers? (ii) Suppose that each correct answer is awarded 5 marks and each incorrect answer carries a penalty of 1 marks, what is the expected overall mark of the student? Solution : 5.2 Hypergeometric distribution Example 5.3 An urn contains 40 white and 60 black balls. 10 balls are selected with replace ment . Let X be the white balls selected. What is the distribution of X Solution: X ∼ B (10 ,. 4) What if the balls were selected without replacement? 52 Definition 5.2 Suppose that n balls are “randomly drawn” (without replacement) from an urn containing N balls, of which k are white and N k are black. Let X be number of white balls selected . Then h ( x ; N,n,k ) = P ( X = x ) = ( k x )( N k n x ) ( N n ) , x = 0 , 1 ,...,n and X is called a hypergeometric r.v. with parameters ( n,N,M ) , denoted by X ∼ H ( n,N,k ) . • If the n balls are chosen randomly with replacement, then X has binomial distribution B ( n,m/N ) • The simplest way to view the distinction between the binomial distribution and the hypergeometric distribution lies in the way the balls are chosen. Hypergeometric distribution – without replacement Binomial distribution – with replacement • The mean and variance of the hypergeometric distribution are μ = n k N , σ 2 = N n N 1 × n k N (1 k N ) • Binomial approximation to the hypergeometric distribution: Intuitively, when N and k are large in relation to n , it shouldn’t make much difference whether the selection is being done with or without replacement. Ifwhether the selection is being done with or without replacement....
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 Fall '09
 lisiufeng
 Math, Statistics, Bernoulli, Binomial, Normal Distribution, Poisson Distribution, Probability, Probability theory, Binomial distribution

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