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Stat447F08HW6Sol - Prepared by Chuanlong Du Oct 22,2008 1...

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Unformatted text preview: Prepared by Chuanlong Du Oct 22,2008 1. Ex5.34 From Theorem 5.4 in the text book we know that the moment generating function of the binomial distribution is ? ¡ ?¢ = £ 1 + ? ? − 1 ¢¤ ? lim ?→∞ , → ? = ¥ ? ¡ ?¢ = lim ?→∞ , → ? = ¥ £ 1 + ? ? − 1 ¢¤ ? = lim ?→∞ , → ? = ¥ ¦ 1 + ? ? ? − 1 ¢ ? § ? = lim ?→∞ ¦ 1 + ¥ ? ? − 1 ¢ ? § ? = ? ¥¨? ? − 1 © which is the moment generating function of the Poisson distribution. 2. Ex5.62 The negative binomial random variable is the number of trials on which the ª ?? success occurs, and the geometric random variable is the number of trials on which the 1 st success occurs. Thus we know that a geometric distribution with parameter is the negative binomial distribution with parameters 1 and . So we have « ¬ ; ¢ = ­ ∗ ¬ ; 1, ¢ = 1 ¬ ­ 1; ¬ , ¢ (b) Refer to Table one, we find that ­ 1; 6, 0.3 ¢ = 0.3025 . So we know that « 6; 0.3 ¢ = 1 6 ­ 1; 6, 0.3 ¢ = 1 6 × 0.3025 = 0.0504 3. Ex5.67 (a) ? ® = 12 200 = 0.06 > 0.05 , not satisfied (b) ?...
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