Massachusetts Institute of Technology Department of Electrical Engineering & Computer Science 6.041/6.431: Probabilistic Systems Analysis (Spring 2006) 1. s 3 2 1 0 (12/90) (18/90) (12/90) (18/90) (6/90) (8/90) (12/90) (4/90) Tutorial 3 March 2-3, 2006 The joint PMF for r andom variables R and S is depicted in the sketch as follows: A point at (r, s) is labeled with P(R = r, S = s) for all pairs with positive probability. Let A denote the event {S �= 3}. (a) Prepare neat, fully-labeled sketches of pS (s) and pS|A(s). (b) Let Y = R − S. Prepare a neat, fully-labeled sketch of pR,Y (r, y). (c) Define th e random variable X = R + S. Prepare a neat, 0 1 2 3 r fully-labeled plot of pX|A(x). 2. Chuck will go shopping for probability books for K hours. Here, K is a random variable and is equally likely to be 1, 2, 3, or 4. The number of books N that he buys is random and depends on how long he shops. We are told that 1 pN|K (n | k) = , for n = 1, . . . , k. k (a) Find the joint PMF of K and N. (b) Find the marginal PMF of N. (c) Find the conditional PMF of K given that N = 2. (d) We are now told that he bought at least 2 but no more than 3 books. Find the conditional mean and variance of K, given this p iece of information. (e) The cost of each book is a random variable with mean 3. What is the expected value of his total expenditure? Hint: Condition on events N = 1, . . . , N = 4 and use the total expectation theorem. 3. Consider three rand om variables X, Y , and Z, associated with the same experiment. The random variable X is geometric with parameter p. If X is even, then Y and Z are equal to zero. If X is odd, (Y, Z) is un iform ly distribu ted on the set S = {(0, 0), (0, 2), (2, 0), (2, 2)}. The figure below shows all the possible values for the triple (X, Y, Z) that have X ≤ 8. (Note that the X axis starts at 1 and that a complete figure would extend indefinitely to the right.) Page 1 of 2Massachusetts Institute of Technology Department of Electrical Engineering & Computer Science 6.041/6.431: Probabilistic Systems Analysis (Spring 2006) Y 1 2 3 4 5 6 2 0 2 7 8 X (a) (b) Z (c) Find var((Y + Z) | X = 5). Find the joint PMF pX,Y,Z (x, y, z). Answer with “yes” or “no” and one sentence of explanation: (i) Are Y and Z independent? (ii) Given that Z = 2, are X and Y in dependent? (iii) Given that Z = 0, are X and Y in dependent? (iv) Given that Z = 2, are X and Z independent? Page 2 of
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