Hw07 - STAT 410 Homework#7(due Friday March 7 by 3:00 p.m...

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STAT 410 Spring 2008 Homework #7 (due Friday, March 7, by 3:00 p.m.) 1. Suppose X has a multivariate normal N 3 ( μ , Σ ) distribution with mean μ = ° ° ° ± ² ³ ³ ³ ´ µ 23 17 7 and covariance matrix Σ = ° ° ° ± ² ³ ³ ³ ´ µ - - 25 10 0 10 9 2 0 2 4 . a) Find P ( X 1 > 10 ) . b) Find P ( X 1 > 10 | X 2 = 20, X 3 = 25 ) . c) Find P ( X 2 < 20 ) . d) Find P ( X 2 < 20 | X 1 = 10, X 3 = 20 ) . e) Find P ( 3 X 1 + 2 X 2 + X 3 > 60 ) . 2. Let X be a random variable with a Gamma distribution with α = 3 and θ = 5 ( i.e., λ = 0.2 ) . Find the probability P ( X > 31.48 ) a) … by integrating the p.d.f. of the Gamma distribution; b) … by using the relationship between Gamma and Poisson distributions; Hint: If X has a Gamma ( α , θ = 1 / λ ) distribution, where α is an integer, then F X ( t ) = P ( X t ) = P ( Y α ) , where Y has a Poisson ( λ t ) distribution. c) … by using the relationship between Gamma and Chi-square distribution. Hint: If X has a Gamma ( α , θ = 1 / λ ) distribution, where α is an integer, then 2 X / θ has a chi-square distribution with 2 α degrees of freedom. See p. 673 of the textbook for the Chi-square Distribution table.
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