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# Hw 8 - Stat 116 Homework 8 Due Wednesday June 4th Please...

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Stat 116 Homework 8 Due Wednesday, June 4th. Please show work and justify answers. No credit for a final answer with no explanation, even if the answer is correct. 1. Prove that Binomial random variables Bin ( n, p ) converge in distribution to a poisson random variable with parameter λ as n tends to infinity, if np = λ (with λ fixed). Solution: The MGF of a Bin ( n, p ) is ( pe t + 1 - p ) n . Now let n → ∞ with np = λ , fixed. Then the MGF become ( pe t + 1 - p ) n = λ n e t + 1 - λ n n = 1 + ( e t - 1) λ n n -→ e λ ( e t - 1) as n → ∞ since (1 + x/n ) n e x and so we recognize the above quantity as the MGF of a Poisson ( λ ) r.v. . 2. Let X i , i = 1 , 2 , ..., n be independent Bernoulli( p ) random variables and let Y n = 1 n n i =1 X i . Recall, that if U n converges in distribution to W and V n converges in prob- ability to c then by Slutsky’s Theorem U n V n converges in distribution to cW , you may use this result for this problem. (a) Show that n ( Y n - p ) N [0 , p (1 - p )].

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