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MIT6_041F10_assn06

# MIT6_041F10_assn06 - Massachusetts Institute of Technology...

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Massachusetts Institute of Technology Department of Electrical Engineering & Computer Science 6.041/6.431: Probabilistic Systems Analysis (Fall 2010) Problem Set 6 Due October 27, 2010 1. Random variables X and Y are distributed according to the joint PDF ax, if 1 x 2 and 0 y x, f X,Y ( x, y ) = 0 , otherwise . (a) Evaluate the constant a . (b) Determine the marginal PDF f Y ( y ). (c) Determine the conditional expectation of 1 /X given that Y = 3 / 2. (d) Random variable Z is defined by Z = Y X . Determine the PDF f Z ( z ). 2. Let X and Y be two independent random variables. Their probability densities functions are shown below. 1 0.8 f X ( x ) 0.6 0.4 0.2 0 x 1 4 f X ( x ) = 3 (1 x 2 ) −1.5 −1 −0.5 0 0.5 1 1.5 f Y ( y ) 0.8 0.6 0.4 0.2 0 1 3 3 2 0 0.5 1 1.5 2 2.5 3 y Let Z = X + Y . Determine f Z ( z ). 3. Consider n independent tosses of a k -sided fair die. Let X i be the number of tosses that result in i . (a) Are X 1 and X 2 uncorrelated, positively correlated, or negatively correlated? Give a one-line justification.

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