ps8 - ECE 804, Random Signal Analysis Nov. 23, 2009 OSU,...

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Unformatted text preview: ECE 804, Random Signal Analysis Nov. 23, 2009 OSU, Autumn 2009 Due: Dec. 2, 2009 Problem Set 8 Problem 1 Let the Gaussian random variable X N (0 , 1) and the Bernoulli random variable Z = ( 1 , with prob. 1/2- 1 , with prob. 1/2 be independent. Define Y = ZX . (a) Find f Y ( y ). (b) Show that X and Y are uncorrelated. (c) Find f Y | X ( y | x ) and f X,Y ( x,y ). (d) Are X and Y independent? Are they jointly Gaussian? Explain. Problem 2 Let X and Y be jointly uniform in the region enclosed by X > , Y > 0 and Y 6 2- 2 X . (a) Find f X ( x ) and f Y ( y ). (b) Find f Y ( y | X = x ). (c) If we observe that Y = 1, what is the expected value and the variance of X ? Problem 3 The random variable X is uniformly distributed in the interval between 0 and 1; the random variable Y is then selected from the interval between 0 and x with a pdf, f Y | X ( y | x ), that is symmetric about x/ 2. An example of a possible pdf is shown below....
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ps8 - ECE 804, Random Signal Analysis Nov. 23, 2009 OSU,...

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