6.436J Fundamentals of Probability, Problem Set 8
6.436J Fundamentals of Probability, Problem Set 8
6.436J Fundamentals of Probability, Problem Set 8
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MASSACHUSETTS INSTITUTE OF TECHNOLOGY<strong>6.436J</strong>/15.085J Fall 2008<strong>Problem</strong> <strong>Set</strong> 8 due 11/12/2008Readings: Notes for Lectures 14-16.Optional Readings:[GS] Section 4.9 (multivariate normal)[GS] Sections 5.7-5.8 (characteristic functions)Exercise 1. Let X be a random variable with mean, variance, and moment generatingfunction), denoted by E[X], var(X), and M X (s), respectively. Similarly,let Y be a random variable associated with E[X], var(X), and M X (s).Each part <strong>of</strong> this problem introduces a new random variable Q, H, G, D. Determinethe means and variances <strong>of</strong> the new random variables, in terms <strong>of</strong> themeans, and variances <strong>of</strong> X and Y .(a) M Q (s) = [M X (s)] 5 .(b) M H (s) = [M X (s)] 3 [M Y (s)] 2 .(c) M G (s) = e 6s M X (s).(d) M D (s) = M X (6s).Exercise 2. A random (nonnegative integer) number <strong>of</strong> people K, enter a restaurantwith n tables. Each person is equally likely to sit on any one <strong>of</strong> the tables,independently <strong>of</strong> where the others are sitting. Give a formula, in terms <strong>of</strong> themoment generating function M K (·), for the expected number <strong>of</strong> occupied tables(i.e., tables with at least one customer).Exercise 3. Suppose thatlim sup log P(X > x) −ν < 0. (1)x→∞ xEstablish that M X (s) < ∞ for all s ∈ [0, ν).Exercise 4. Suppose that X, Z 1 , . . . , Z n have a multivariate normal distribution,and X has zero mean. Furthermore,suppose that Z 1 , . . . , Z n are independent.nShow that E[X | Z 1 , . . . , Z n ] =i=1 E[X | Z i].1
MIT OpenCourseWarehttp://ocw.mit.edu<strong>6.436J</strong> / 15.085J <strong>Fundamentals</strong> <strong>of</strong> <strong>Probability</strong>Fall 2008For information about citing these materials or our Terms <strong>of</strong> Use, visit: http://ocw.mit.edu/terms.