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

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

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

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

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 3

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

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

View Full Document
Ask a homework question - tutors are online