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homework10

# homework10 - Z 2 5(a X and Y are independent and...

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1 EL 630: Homework 10 1. The correlation coefficient of two random variables X and Y is defined as [] ) ( ) ( ) )( ( Y Var X Var Y X E Y X XY μ ρ - - = Show that . 1 | | XY 2. X and Y are zero mean independent r.vs with variance 2 1 σ and 2 2 respectively, i.e., X ); , 0 ( 2 1 N Y ). , 0 ( 2 2 N Let . 0 , + + = c c bY aX Z (a) Find the characteristic function ) ( u Z Φ of Z . (b) Using ) ( u Z Φ conclude that Z is also a Normal r.v. (c) Use (b) to find the mean and variance of Z . 3. The random variables X and Y are jointly distributed over the region 1 0 < < < y x as < < < = , , 0 , 1 0 , ) , ( otherwise y x kx y x f XY for some k . (Find k ). Also find the variances of X and Y . What is the covariance between X and Y . 4. X is a Poisson r.v with parameter λ and Y is a normal r.v with mean and variance . 2 Further X and Y are given to be independent. (a) Find the joint characteristic function of X and Y . (b) Define . Y X Z + = Find the characteristic function of

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Unformatted text preview: Z . 2 5. (a) X and Y are independent and identically distributed normal random variables with mean 0 and variance . 2 σ . 2 1 ) ( ) ( ) , ( 2 2 2 2 / ) ( 2 πσ y x Y X XY e y f x f y x f +-= = Define . 2 , 2 2 2 2 2 2 Y X XY V Y X Y X U + = +-= Find the joint p.d.f ) , ( v u f UV of the random variables U and V . Using the expression for ) , ( v u f UV conclude whether U and V are independent or not. (b) X and Y are independent exponential r.vs with common parameter . λ Find (i) [ ] ) , min( Y X E (ii) [ ] . ) , 2 max( Y X E ( Hint: Use the fact that [ ] ∫∫ = dxdy y x f y x g Y X g E XY ) , ( ) , ( ) , ( )....
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homework10 - Z 2 5(a X and Y are independent and...

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