PS-5-2009

PS-5-2009 - EE 261 The Fourier Transform and its...

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EE 261 The Fourier Transform and its Applications Fall 2009 Problem Set Five Due Wednesday, October 28 1. (10 points) Expected values of random variables, orthogonality, and approximation Let X be a random variable with probability distribution function p ( x ). Recall that the mean, or expected value, of X is the number E ( X ) = integraldisplay −∞ xp ( x ) dx. Some important properties of the expected value (not trivial) are: Linearity E ( αX 1 + βX 2 ) = α E ( X 1 ) + β E ( X 2 ) [Note: there is no assumption that X 1 and X 2 are independent. The additivity of expected values when the random variables are not independent has been called the First Fundamental Mystery of Probability.] Functional dependence If Y = f ( X ) for a function f then E ( Y ) = integraldisplay −∞ f ( x ) p ( x ) dx, where p ( x ) is the probability distribution function of X . Let R 0 be the collection of random variables with expected value 0 and finite variance. There is a close analogy between the inner product of two vectors (including functions) and the expected value of the product of two random variables in R 0 , namely one sets ( X 1 ,X 2 ) = E ( X 1 X 2 ) , the expected value of the product of X 1 ,X 2 ∈ R 0 . Two random variables in R 0 are orthogonal if ( X 1 ,X 2 ) = 0. If X 1 and X 2 are independent then E ( X 1 X 2 ) = E ( X 1 ) E ( X 2 )

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