# hw4_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, [email protected] Homework 4 - Solutions Problem 1 The runner-up in a road race is given a reward that depends on the difference between his time and the winner’s time. He is given 50 dollars for being zero to two minutes behind, 30 dollars for being two to ﬁve minutes behind, 10 dollars for being 5 to 10 minutes behind, and nothing otherwise. Given that the difference between his time and the winner’s time is uniformly distributed between 0 and 15 minutes, ﬁnd the mean and variance of the reward of the runner-up. Solution : Let X be the reward. The probability that X = 50 is 2 15 , the probability that X = 30 is 3 15 , and the probability that X = 10 is 5 15 . Therefore E [ X ] = 2 15 · 50 + 3 15 · 30 + 5 15 · 10 = 16 , and var ( X ) = E [ X 2 ] - ( E [ X ]) 2 = 2 15 · 50 2 + 3 15 · 30 2 + 5 15 · 10 2 - (16) 2 = 290 . 6667 . Problem 2 Let X be a random variable with PDF f X ( x ) = ± 3 x 2 7 if 1 < x 2 0 otherwise. and let Y = X 2 . Calculate E [ Y ] and var ( Y ) . Solution: We have E [ Y ] = E [ X 2 ] = Z 2 1 x 2 f X ( x ) dx = Z 2 1 3 x 4 7 dx = 3 5 x 5 7 ² ² ² 2 1 = 3 5 ( 32 7 - 1 7 ) = 93 35 = 2 . 6571 To obtain the variance of Y , we ﬁrst calculate E [ Y 2 ] . We have, after straightforward calculation, E [ Y 2 ] = E [ X 4 ] = Z 2 1 x 4 f X ( x ) dx = Z 2 1 3 x 6

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.

## 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.

### Page1 / 4

hw4_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