zz-11 - ( x , 2 x ). 6. Suppose X N (0 , 1), and dene Y = X...

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Utah State University ECE 6010 Stochastic Processes Homework # 2 Due Friday Sept. 16, 2005 1. Suppose X is a r.v. with c.d.f. F X . Prove the following: (a) F X is nondecreasing. (b) lim a →∞ F X ( a ) = 1. (c) lim a →-∞ F X ( a ) = 0. (d) F X is right continuous. (e) P ( a < X b ) = F X ( b ) - F X ( a ) if b > a . (f) P ( X = a ) = F X ( a ) - lim b a - F X ( b ) . Also, ±nd expressions for P ( a X b ), P ( a X < b ) and P ( a < X < b ) in terms of F X . 2. Show that the following are valid p.m.f.s: (a) Binomial: f X ( a ) = n ! / (( n - a )! a !) π a (1 - π ) n - a if a ∈ { 0 , 1 , . . . , n } . (b) Poisson: f X ( a ) = e - λ λ a /a ! for a ∈ { 0 , 1 , . . . } . 3. Find the mean and variance of X when X is (a) N ( μ, σ 2 ); (b) Binomial( n, π ); (c) Poisson( λ ); (d) Exponential( λ ). Do not use characteristic functions. 4. Suppose that X and Y are jointly continuous. Show that f X ( x ) = Z -∞ f XY ( x, y ) dy x R . 5. Suppose that X and Y are jointly Gaussian with parameters μ x , σ 2 x , μ y , σ 2 y , ρ . Show that X ∼ N
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Unformatted text preview: ( x , 2 x ). 6. Suppose X N (0 , 1), and dene Y = X 2 . Are X and Y uncorrelelated? Are X and Y independent? Find the p.d.f. of Y . Are X and Y jointly continuous? 7. Show that cov( aX + b, cY + d ) = ac cov( X, Y ). 1 8. Suppose X N (0 , 2 ). Use the ch.f. of X to Fnd an expression for E [ X n ], n Z + . Problems from the Grimmett &amp; Stirzaker text: 1. Ex 1.4.4 2. Ex 1.4.5. Hint: Let C i be the event that the i th door conceals the car, let G be the event that you see a goat, and let B be the event that you see Bill. 3. Ex 1.5.1. 4. Ex 1.5.2 5. Ex 1.5.7. 6. Prob. 1.8.5 7. Prob. 1.8.6. 8. Prob. 1.8.19. 9. Prob. 1.8.20. 10. Prob. 1.8.30. 2...
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zz-11 - ( x , 2 x ). 6. Suppose X N (0 , 1), and dene Y = X...

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