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Hw8Sol - SIEO 3600(IEOR Majors Introduction to Probability...

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SIEO 3600 (IEOR Majors) Solution to Assignment #8 Introduction to Probability and Statistics March 31, 2010 Solution to Assignment #8 ( Binomial, Geometric and Uniform random variables ) 1. Polling: During a Presidential election (only two candidates for simplicity). (a) It has been determined (by extensive sample) that p = 55% of all voters will vote for candidate A and q = 1 - p = 45% for candidate B . If you randomly select 5 voters, what is the probability that exactly 3 will vote for candidate A ? (b) Suppose that apriori we do not know the proportion p of voters who will vote for can- didate A , but that when we randomly selected 5 voters, exactly 3 said they would vote for A . Find the value of p that maximizes the probability of this event (e.g., of getting exactly 3 out the 5 saying they would vote for A .) Solution: (a) Let X be the number of voters who vote for A . Then X is a binomial random variable with parameter p = 0 . 55 and n = 5. Therefore, P ( X = 3) = 5 3 0 . 55 3 0 . 45 2 . (b) We know that X is a geometric random variable with n = 5 but unknown p . Also, we have P ( X = 3) = 5 3 p 3 (1 - p ) 2 . (1) We want to maximize P ( X = 3) such that 0 < p < 1. Taking the derivative of (1) with respect to p and let it be 0, we have d dp P ( X = 3) = 5 3 ( 3 p 2 (1 - p ) 2 + p 3 2(1 - p )( - 1) ) = 0 , (2) implies that p * = 3 / 5 is the maximizer. 2. Let X be a binomial random variable with E [ X ] = 7 and var( X ) = 2 . 1 Find (a) P ( X = 4); (b) P ( X > 12).
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2 SIEO 3600, Solution to Assignment #8 Solutions: (a) Because X follows a binomial Bino ( n, p ), therefore E [ X ] = np = 7 var( X ) = np (1 - p ) = 2 . 1 n = 10 p = 0 . 7 Hence, P ( X = 4) = ( 10 4 ) 0 . 7 4 0 . 3 6 . (b) Since n = 10, the state space of X is S X = { 0 , 1 , 2 , . . . , 10 } , so P ( X > 12) = 0. 3. Suppose that a particular trait (such as eye color or left-handedness) of a person is classified on the basis of one pair of genes, and suppose that d represents a dominant gene and r a recessive gene. Thus, a person with dd genes is pure dominance, one with rr is pure recessive, and one with rd is hybrid. The pure dominance and the hybrid are alike in appearance.
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