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Mathematical Statistics Quiz#1-SOL--2009

Mathematical Statistics Quiz#1-SOL--2009 - Mathematical...

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Unformatted text preview: Mathematical Statistics Quiz #1 4/16/2009 10:00‐12:00 1. Prove the following statements (1) If events A and B are independent, then A and B, A and B are independent. SOL: (2) If P A|B SOL: , then P B|A . PA B APB 2. A random number of fair dice is thrown, where , 1, , . Let denote the sum of the scores on the dice. Find the probability that (1) 2, given 3. SOL: By Bayes’ rule P N 2S 3 3 1 2 3 k PS |N PS PN 3|N kPN k Where P S But P S P S P S P S 3N 3|N 3|N 3|N ∑ P P PS 1,2 , 2,1 1,1,1 4 0, k P S 3 ∑ 2S PS 3 3|N PS kPN |N PS PN k P N (2) 3 given is odd. PN 1 PS PN 3|N 3 1PN N SOL: P N is odd P S 3 1 PS 3|N 3PN 3 N is odd P S 3|N is odd PS PN 3. Let X and Y be independent random variables which have the same distribution 1 (1) Calculate SOL: ∑ . ∑ 1 , , 0,1, ∑ . (2) Show that SOL: , , ∑ , 1 1 ∑ 0,1, 1 , ∑ ∑ 1 1 4. Suppose n cards numbered 1,2, , n are laid out at random in a row. We say a match occurs at position i if the ith card appears in the ith position. Let X 1 if there is a match position I and X 0 otherwise. Let S denote the number of matches then S X X X. (a) Compute E X , V X , and COV X , X . SOL: E X PX 1 ! ! VX COV X , X ! ! (b) Show that E S SOL: E S V S EX ∑ 1 and V S X X n 1 n 2∑ 1 n 1 1 n X 1. n 1 COV X , X 2 1 n 2 1 nn 1 1 1 n nn 1 n 5. Let X be a random variable with moment generating function MX t E eX e σ . !σ ! σ ! σ ! Show that E X SOL: e σ 0 and E X σ ! σ . Find E X σ and EX . 1 1 1 0 t t EX ! 0 t t EX ! σ ! t t EX ! σ ! t EX ! tE X t t Comparing the coefficients, we have EX ! ! σ , i.e., E X !σ ! and E X 0 6. Consider the following joint probability density function (pdf) f x, y λe 0, λ ,0 x y elsewhere ∞, λ 0 (a) Find the marginal pdfs fX x and fY y . (b) Find the conditional pdfs fX| x|y and fY|X x|y . (c) Find E X|Y SOL: (a) fX x y. λe λ dy λe λ ,x 0 fY y (b) fX|Y x|y λe , Y λ dx λ λ λ λ λ ye λ ,y x λe 0 y λ ,0 λe λe λ λ ∞ ,0 x y ∞, λ 0 fY|X x|y (c) E X|Y y f x, y fX y ...
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