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Unformatted text preview: Massachusetts Institute of Technology Department of Electrical Engineering & Computer Science 6.041/6.431: Probabilistic Systems Analysis (Fall 2010) Problem Set 5 Due October 18, 2010 1. Random variables X and Y are distributed according to the joint PDF ax, if 1 ≤ x ≤ y ≤ 2 , f X,Y ( x,y ) = , otherwise. (a) Evaluate the constant a . (b) Determine the marginal PDF f Y ( y ). (c) Determine the expected value of 1 X , given that Y = 3 2 . 2. Paul is vacationing in Monte Carlo. The amount X (in dollars) he takes to the casino each evening is a random variable with the PDF shown in the figure. At the end of each night, the amount Y that he has on leaving the casino is uniformly distributed between zero and twice the amount he took in. f X (x ) (a) Determine the joint PDF f X,Y ( x,y ). Be sure to indicate what the sample space is. (b) What is the probability that on any given night Paul makes a positive profit at the casino? Justify your reasoning. (c) Find and sketch the probability density function of Paul’s profit on any particular night, Z = Y − X . What is E [ Z ]? Please label all axes on your sketch. 40 x (dollars) Page 1 of 3 Massachusetts Institute of Technology Department of Electrical Engineering & Computer Science 6.041/6.431: Probabilistic Systems Analysis (Fall 2010) 3. X and Y are continuous random variables. X takes on values between 0 and 2 while Y takes on values between 0 and 1. Their joint pdf is indicated below. values between 0 and 1....
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 Fall '10
 Prof.DimitriBertsekas
 Electrical Engineering, Probability theory, probability density function, Probabilistic Systems Analysis, symb ol, indep endent

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