6.436J Fundamentals of Probability, Problem Set 8

MASSACHUSETTS INSTITUTE OF TECHNOLOGY6.436J/15.085J Fall 2008Problem Set 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 of this problem introduces a new random variable Q, H, G, D. Determinethe means and variances of the new random variables, in terms of themeans, and variances of 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 of people K, enter a restaurantwith n tables. Each person is equally likely to sit on any one of the tables,independently of where the others are sitting. Give a formula, in terms of themoment generating function M K (·), for the expected number of 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.edu6.436J / 15.085J Fundamentals of ProbabilityFall 2008For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.