Mathematical Statistics Quiz#1--2009

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

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Unformatted text preview: Mathematical Statistics Quiz #1 4/16/2009 1. Prove the following statements (1) If events A and B are independent, then A′ and B, A′ and B′ are independent. (2) If P A|B , then P B|A . 2. A random number of fair dice is thrown, where , Let (1) (2) 1, ,. denote the sum of the scores on the dice. Find the probability that 2, given 3. 3 given is odd. 3. Let X and Y be independent random variables which have the same distribution 1 (1) Calculate (2) Show that . , 0,1, ,. , 0,1, . 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 . (b) Show that E S 1 and V S 1. 5. Let X be a random variable with moment generating function MX t E eX e σ . !σ ! Show that E X 0 and E X . Find E X and EX . 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 y. ...
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