Massachusetts Institute of TechnologyDepartment of Electrical Engineering & Computer Science6.041/6.431: Probabilistic Systems Analysis(Fall 2010)Problem Set 6Due October 27, 20101. Random variables X and Y are distributed according to the joint PDFfX,Y(x, y) =ax, if 1 ≤ x ≤ 2 and 0 ≤ y ≤ x,0, otherwise.(a) Evaluate the constant a.(b) Determine the marginal PDF fY(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 fZ(z).2. Let X and Y be two independent random variables. Their probability densities functions areshown below.−1.5 −1 −0.5 0 0.5 1 1.500.20.40.60.81xfX(x)fX(x) =34(1 − x2)0 0.5 1 1.5 2 2.5 300.20.40.60.81yfY(y)1323Let Z = X + Y . Determine fZ(z).3. Consider n in dependent tosses of a k-sided fair die. Let Xibe the number of tosses that resultin i.(a) Are X1and X2uncorrelated, positively correlated, or negatively correlated? Give a one-linejustification.(b) Compute the covariance cov(X1, X2) of X1and X2.Page 1 of 2Massachusetts Institute of TechnologyDepartment of Electrical Engineering & Computer Science6.041/6.431: Probabilistic Systems Analysis(Fall 2010)4. Random variables X and Y have the joint PDF shown below:yx1.0 2.0-1.0-1.01.02.0f (x,y) = 0.1 X,Y- 2.0(a) Find the conditional PDFs fY |X(y | x) and fX|Y(x | y), for various values of x and y,respectively.(b) Find E[X | Y = y], E[X], and var(X | Y = y). Use these to calculate var(X).(c) Find E[Y | X = x], E[Y ], an d var(Y | X = x). Use these to calculate var(Y ).5. The wombat club has N members, where N is a random variable with PMFpN(n) = pn−1(1 − p) for n = 1, 2, 3, . . ..On the second Tu esday night of every month, the club holds a meeting. Each wombat memberattends the meeting with probability q, independently of all the other members. If a wombatattends the meeting, then it brings an amount of money, M, which is a continuous randomvariable with PDFfM(m) = λe−λmfor m ≥ 0.N, M, and whether each wombat member attends are all independent. Determine:(a) The expectation and variance of the number of wombats showing up to the meeting.(b) The expectation and variance for the total amount of money brought to the meeting.G1†. (a) Let X1, X2, . . . , Xn, Xn+1, . . . , X2nbe independent and identically d istributed random vari-ables.FindE[X1| X1+ X2+ . . . + Xn= x0],where x0is a constant.(b) DefineSk= X1+ X2+ . . . + Xk, 1 ≤ k ≤ 2n.FindE[X1| Sn= sn, Sn+1= sn+1, . . . , S2n= s2n],where sn, sn+1, . . . , s2nare constants.†Required for 6.431; optional for 6.041 Page 2 of
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