hw4 - f X,Y ( x,y ) = 1 2 1 2 p 1- 2 exp -1 2(1- 2 )...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
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
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 2
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: f X,Y ( x,y ) = 1 2 1 2 p 1- 2 exp -1 2(1- 2 ) &quot; x-m 1 1 2-2 x-m 1 1 y-m 2 2 + y-m 2 2 2 # for- &lt; x &lt; and- &lt; y &lt; . Find P [ X 2 + Y 2 &lt; R 2 ] when = 0, m 1 = m 2 = 0, and 1 = 2 = . 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 2 , y 2 (a) Find the value of the constant c . (b) Find the joint cdf of X and Y . (c) Find the marginal pdfs of X and Y . Problem 9. Let ( X,Y ) have the joint pdf f X,Y ( x,y ) = k ( x + y ) &lt; x &lt; 1 , &lt; y &lt; 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 &lt; otherwise Evaluate the conditional expectations E [ X | y ] and E [ Y | x ]. 2...
View Full Document

Page1 / 2

hw4 - f X,Y ( x,y ) = 1 2 1 2 p 1- 2 exp -1 2(1- 2 )...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online