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assignment7 - EE 505 B Fall 2011 Assignment 7 For full...

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EE 505 B Fall 2011 Assignment 7 For full credit, you must show all of the steps or reasoning that you used to get the answer. 1. Let X be a random variable with a uniform probability density function. f X ( x ) = braceleftBigg 1 b - a 0 < a < x < b 0 otherwise (a) Find E { X } . Let Y = X 2 (b) Using the equation E { Y } = integraldisplay - yf Y ( y ) dy find E { Y } . (c) Using the fact that E { Y } = E { X 2 } , find E { Y } . 2. (a) Let the random variable Y = a cos ( ω t + Θ ) where a , ω , and t are constants, and Θ is a uniform random variable in the interval [ 0, 2 π ) . (One can imagine that the random variable Y results from sampling the amplitude of a sinusoid that has random phase Θ .) Find the expected value of Y , the expected value of the power in Y , i.e., E { Y 2 } , and the variance of Y . (b) Let the random variable Y = A cos ( ω t )+ c , where A has mean m X and vari- ance σ 2 X , and ω and c are constants. Find the mean and variance of Y . 3. Find the expected value of the received signal X in the detection theory example from class. Hint: the events, 0 T and 1 T
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