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hw4sol - STATS 116 Kihwan Choi Homework 4 solutions 1(3.3.9...

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STATS 116 Kihwan Choi Homework 4 solutions 1. (3.3.9) Out of n individual voters at an election, r vote Republican and n - r vote Democrat. At the next election the probability of a Republican switching to vote Democrat is p 1 , and of a Democrat switching is p 2 . Suppose individuals behave independently. Find (a) the expectation and (b) the variance of the number of Republican votes at the second election. Solution: (a) Let X denote the number of Republican votes at the second election. Also X 1 and X 2 denote the number of Republican votes at the second election with Republican votes at the first election and the number of Republican votes at the second election with Democrat votes at the first election, respectively. Then X = X 1 + X 2 where X 1 B ( r, 1 - p 1 ) and X 2 B ( n - r, p 2 ). Thus, EX = EX 1 + EX 2 = r (1 - p 1 ) + ( n - r ) p 2 (b) Since X 1 and X 2 are independent, V ar ( X ) = V ar ( X 1 ) + V ar ( X 2 ) = rp 1 q 1 + ( n - r ) p 2 q 2 2. (3.3.12) A random variable X has expectation 10 and standard deviation 5. (a) Find the smallest upper bound you can for P ( X 20). Solution: By one-sided Chebyshev’s inequality (Cantelli’s inequality), P ( X 20) = P ( X - μ σ 2) 1 1 + 2 2 = 0 . 2 , which can be achievable when P ( X ) = 0 . 2 for X = 20 , 0 . 8 for X = 7.5 , resulting in μ = 10 and σ = 5. (b) Could X be a binomial random variable? Solution: No, it is impossible. If it is, we should have np = μ = 10 and npq = σ 2 = 25. This implies q = 2 . 5, but q = 1 - p should be in [0 1].
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