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# homework1 - Stochastic Processes nctuee07f Homework 1 Due...

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Stochastic Processes nctuee07f Homework 1 Due on Friday, 10/12/2007 , 1:00 PM at EC 713 1. Let X be a continuous random variable, and Y a discrete random variable. Show that the pdf f X ( x ) can be expressed by f X ( x ) = X n f X | Y ( x | Y = n ) P [ Y = n ] . (Hint: Use f X ( x ) = d dx P [ X x ] . ) 2. A binary signal S ∈ {- 1 , +1 } is transmitted, and we are given that P ( S = 1) = P ( S = - 1) = 1 / 2. The received signal at the receiver is modeled by Y = S + N, where N is Gaussian noise, with zero mean and variance σ 2 , independent of S . What is the probability that S = - 1, given that we have observed Y = y ? 3. (a) Show that E [ E [ X | Y ]] = E [ X ] . (b) Let X and Y be two random variables, either discrete or continuous. Show that E h g ( X ) · h ( Y ) Y = y i = h ( y ) · E h g ( X ) Y = y i for any real-valued functions g and h . (c) The conditional variance of X given Y = y is deﬁned by Var( X | Y = y ) , E £ ( X - E [ X | Y = y ]) 2 | Y = y / , it is clear that Var(
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