EE465-USC-Fall08-Assignment3

# EE465-USC-Fall08-Assignment3 - X 2 E Y 2(1(Hint Use the...

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EE 465 : Homework 3 Due : 09/23/2008, Tuesday in class. 1. Let X be a geometric random variable. Find and plot F X ( x | A ) if: a) A = { X > k } where k is a positive integer b) A = { X < k } where k is positive integer 2. Let X = ( x 1 x 2 x 3 ) T be a Gaussian random vector and X η ( μ X , σ X ) where μ X = (1 5 2) T and σ X = 1 1 0 1 4 0 0 0 9 a) What are the pdfs of i) x 1 ii) x 2 + x 3 iii) 2 x 1 + x 2 + x 3 b) What is P (2 x 1 + x 2 + x 3 < 0)? c) What is the pdf of Y = AX where A = p 2 1 1 1 - 1 1 P 3. Let X and Y be jointly gaussian random variables with zero mean, variance σ 2 X = 2 and σ 2 Y = 4, and normalized correlation coe±cient ρ XY = 0 . 5. a) Find the pdf of X | Y = 1. b) Let Z = 2 X + Y + 2. Find E [ Z | Y = 1]. 4. Schwartz Inequality : Prove the following inequality: ( E ( XY )) 2 E (

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Unformatted text preview: X 2 ) E ( Y 2 ) (1) (Hint: Use the fact that for any real a , E (( X + aY ) 2 ) ≥ 0.) 5. Two continuous random variables X and Y are described by the pdf f XY ( x,y ) = c, if | x | + | y |≤ 1 √ 2 , otherwise (2) where c is a constant. a) Find c . b) Find f X ( x ) and f X | Y ( x | y ) . c) Are X and Y independent random variables? Justify your answer. Problem d is an optional question, those who solve it will get extra credits. d) De±ne the random variable Z = ( | X | + | Y | ). Find the pdf f Z ( z ). 2...
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EE465-USC-Fall08-Assignment3 - X 2 E Y 2(1(Hint Use the...

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