hw5_sol - EE 351K Probability, Statistics, and Random...

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EE 351K Probability, Statistics, and Random Processes SPRING 2009 Instructor: Shakkottai/Vishwanath { shakkott,sriram } @ece.utexas.edu Homework 5 - Solutions Problem 1 Consider two continuous random variables Y and Z and a random variable X that is equal to Y with probability p and to Z with prbability 1 - p . Show that the PDF of X is given by f X ( x ) = pf Y ( x ) + (1 - p ) f Z ( x ) Solution : f X ( x ) = lim Δ 0 P ( x<X x +Δ) Δ = lim Δ 0 P ( X = Y ) × P ( x<Y x +Δ)+ P ( X = Z ) × P ( x<Z x +Δ) Δ = p × lim Δ 0 P ( x<Y x +Δ) Δ + (1 - p ) × lim Δ 0 P ( x<Z x +Δ) Δ = pf Y ( x ) + (1 - p ) f Z ( x ) Problem 2 The random variable X has the PDF f X ( x ) = ± cx - 2 , 1 x 4 0 , otherwise 1. Determine the value of c . 2. Let A be the event { X > 2 } . Calculate P ( A ) and the conditional PDF of X given that event A has occured. Solution : Part 1: We know that R -∞ f X ( x ) dx = 1. Hence, R 4 1 c x 2 dx = 1 . c × (1 - 1 4 ) = 1 . c
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This note was uploaded on 10/26/2009 for the course EE 351k taught by Professor Bard during the Spring '07 term at University of Texas.

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hw5_sol - EE 351K Probability, Statistics, and Random...

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