HW14 - University of Illinois Fall 2009 ECE 313: Problem...

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University of Illinois Fall 2009 ECE 313: Problem Set 14: Solutions Conditional Distributions; Covariance and Correlation 1. [Is gambling any good?] (a) Since f X ( · ) is a PDF, it has to integrate to 1 over the real line. Mathematically this means that C 2 10 4 = 1, so C = 2 · 10 - 4 . (b) The amount John returns with is uniformly distributed between 0 and 2 α . So, f Y | X ( v | α ) = ( 1 2 α , 0 v 2 α, 0 , otherwise . (c) Since Y is uniformly distributed between 0 and 2 α , the probability that he returns home with more than α dollars is 1 2 . (d) The average of a uniform random variable over an interval is the simply the mid point of the interval. So, the average money John returns from the casino is α dollars. This means that the average profit he makes is zero dollars. (e) We have f X , Y ( u,v ) = f Y | X ( v | u ) · f X ( u ) = 1 2 u · Cu, 0 v 2 u, 0 u 100 = C 2 , 0 v 2 u, 0 u 100 . So the pair of random variables (
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HW14 - University of Illinois Fall 2009 ECE 313: Problem...

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