# hw4 - f X,Y x,y = 1 2 πσ 1 σ 2 p 1-ρ 2 exp ²-1 2(1-ρ...

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Virginia Tech The Bradley Department of Electrical and Computer Engineering ECE 5605 – Stochastic Signals and Systems – Fall 08 Problem Set 4 Problem 1. The random variable X is Gaussian with mean zero and variance σ 2 . Find E [ X | X > 0] and V AR [ X | X > 0]. Problem 2. The random variable X is Gaussian with mean m and standard deviation σ . Find the mean of Y = cos X. Hint : Use the characteristic function deﬁnition. Hint 2 : In case you forgot, cos θ = 1 2 ( e + e - ) . Problem 3. Find the characteristic function of the uniform random variable in [ - b,b ]. Find the mean of this random variable by applying the moment theorem. Problem 4. Find the mean and variance of a Laplacian random variable by applying the moment theorem. Problem 5. The random variable X has the density f X ( x ) = ± e - x x > 0 0 x < 0 . The function g ( X ) is given by g ( X ) = sin X X . Evaluate the mean value of g ( X ). Hint : g ( X ) = sin X X = 1 2 R 1 - 1 e juX du . Problem 6. Let ( X,Y ) have the joint pdf f X,Y ( x,y ) = xe - x (1+ y ) x > 0 , y > 0 Find the marginal pdf of X and Y . Problem 7. The general form of the joint pdf for two jointly Gaussian random variables is

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Unformatted text preview: f X,Y ( x,y ) = 1 2 πσ 1 σ 2 p 1-ρ 2 exp ²-1 2(1-ρ 2 ) " ³ x-m 1 σ 1 ´ 2-2 ρ ³ x-m 1 σ 1 ´³ y-m 2 σ 2 ´ + ³ y-m 2 σ 2 ´ 2 #µ for-∞ < x < ∞ and-∞ < y < ∞ . Find P [ X 2 + Y 2 < R 2 ] when ρ = 0, m 1 = m 2 = 0, and σ 1 = σ 2 = σ . Hint: Use polar coordinates to compute the integral. 1 Problem 8. The random variables X and Y have joint pdf f X,Y ( x,y ) = c sin ( x + y ) ≤ x ≤ π 2 , ≤ y ≤ π 2 (a) Find the value of the constant c . (b) Find the joint cdf of X and Y . (c) Find the marginal pdf’s of X and Y . Problem 9. Let ( X,Y ) have the joint pdf f X,Y ( x,y ) = k ( x + y ) < x < 1 , < y < 1 (a) Are X and Y independent? (b) Find f Y ( y | x ). Problem 10. The random variables X and Y have joint density f X,Y ( x,y ) = ± e-y ≤ x ≤ y < ∞ otherwise Evaluate the conditional expectations E [ X | y ] and E [ Y | x ]. 2...
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## This note was uploaded on 05/05/2010 for the course ECECS 5605 taught by Professor Dasilver during the Fall '08 term at Virginia Tech.

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hw4 - f X,Y x,y = 1 2 πσ 1 σ 2 p 1-ρ 2 exp ²-1 2(1-ρ...

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