Massachusetts Institute of TechnologyDepartment of Electrical Engineering & Computer Science6.041/6.431: Probabilistic Systems Analysis(Spring 2006)Problem Set 6Due: April 5, 20061. Suppose thatMX(s) =13·11 − s+23·33 − s.What is the PDF of X?2. Find the transform of the random variable X with density function:fX(x) =½pλe−λx+ (1 − p)µe−µxfor x ≥ 00 otherwisewhere p is a constant with 0 ≤ p ≤ 1.3. Consider random variable Z with transformMZ(s) =a − 3ss2− 6s + 8.(a) Find the numerical value for the parameter a.(b) Find P(Z ≥ 0.5).(c) Find E[Z] by using the probability distribution of Z.(d) Find E[Z] by using the transform of Z and without explicity using the probability distri-bution of Z.(e) Find var(Z) by using the probability distribution of Z.(f) Find var(Z) by using the transform of Z and without explicity using the probability distri-bution of Z.4. A coin is tossed repeatedly, heads appearing with probability q on each toss. Let random variableT denote the number of tosses when a run of n consecutive heads has appeared for the first time.(a) Show that the PMF for T can be expressed aspT(k) =0 , k < nqn, k = nÃ∞Xi=k−npT(i)!(1 − q)qn, k ≥ n + 1.(b) Determine the transform MT(s) associated with random variable T .(c) Compute E[T ], the expectation of random variable T .5. This problem is based on an example covered in Monday’s lecture (lecture 11). Let X andN be two independent normal random variables. Say X ∼ N (0, σ2x) and N ∼ N(0, σ2n). LetY = X + N. In lecture we saw that Y is also normal. The lecture slides also prove that theconditional PDF fY |X(y|x) is normal.Prove that for every value of y, the conditional density fX|Y(x|y) is normal.Page 1 of 2Massachusetts Institute of TechnologyDepartment of Electrical Engineering & Computer Science6.041/6.431: Probabilistic Systems Analysis(Spring 2006)6. Four fair 6-sided dice are rolled independently of each other. Let X1be the sum of the numberson the first and second dice, and X2be the sum of the numbers on the third and fourth dice.Convolve the PMFs of the random variables X1and X2to find the probability that the outcomesof the four dice rolls sum to 8.7. Consider two independent random variables X and Y . Let fX(x) = 1 − x/2 for x ∈ [0, 2] and 0otherwise. Let fY(y) = 2 − 2y for y ∈ [0, 1] and 0 otherwise. Give the PDF of W = X + Y .G1†. Let X1, X2, . . . , Xnbe drawn i.i.d from the uniform distribution on [0, 1]. Let Y be the minimumof the Xi, and let Z be the maximum of the Xi. Let W = Y + Z. Compute fW(w), and provethat for all ǫ > 0, limn→∞P(|W − 1| > ǫ) = 0. Thus, for large n, with very high probability Wis close to 1.†Required for 6.431; optional for 6.041 Page 2 of
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