351K EE Probability, Statistics, and Random Processes Instructor:
Shakkottai/Vishwanath Homework 4  Solutions SPRING 2008 shakkott,
[email protected]
Problem 1 The runnerup in a road race is given a reward that depends on the
difference between his time and the winner's time. He is given 20 dollars for being one
minute behind, 10 dollars for being one to two minutes behind, 5 dollars for being 2 to 6
minutes behind, and nothing otherwise. Given that the difference between his time and
the winner's time is uniformly distributed between 0 and 12 minutes, find the mean and
variance of the reward of the runnerup. Solution : Let X be the reward. The probability
that X = 20 is probability that X = 5 is 1 . Therefore 3 1 12 , the probability that X = 10 is
1 12 , and the E[X] = 0.083 20 + 0.083 10 + 0.333 5 = 4.167, and var(X) = E[X 2 ] 
(E[X])2 = 0.083 202 + 0.083 102 + 0.333 52  (4.167)2 = 32.636.
Problem 2 Let X be a random variable with PDF fX (x) = and let Y = X 2 . Calculate E[Y ]
and var(Y ). Solution: We have E[Y ] = E[X 2 ] = 2 2x/5 if 2 < x 3, 0 otherwise. 3 3 x2 fX
(x) dx = 2 2 x4 2x3 dx = 5 5 4 3 3 2 = 2 81 16 (  ) = 6.5 5 4 4 To obtain the variance of Y
, we first calculate E[Y 2 ]. We have, after straightforward calculation, 3 E[Y 2 ] = E[X 4 ]
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 Spring '07
 BARD
 Probability theory, Random Processes Instructor

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