UMD Probability Qualifying Exams/Jan2006Probability

__TOC__ Problem 1 Let be i.i.d. r.v.'s such that and a.s., and let . (a) Find a number such that is a martingale and justify the martingale property. (b) Define . Compute . (c) Compute . Solution (a) Each is clearly -measureable and finite a.s. (Hence ). Therefore we only need to verify the martingale property. That is, we want to show We can assert that exists and is finite since each almost surely. Therefore, in order to make a martingale, we must have . Problem 2 Let be independent Poisson processes with respective parameters , where is an unspecified positive real number. For each , let . Show that does not depend on and find explicitly. Solution First let us find the distribution of : Thus by the chain rule, our random variable has probability density function So then Now integrate the remaining integral by parts letting . We get: Repeat integration by parts another times and we get Problem 3 Let be independent random variables such that (a) Find the characteristic function of . (b) Show that converges in distribution to a non-degenerate random variable. Solution (a) Then by independence, we have Problem 4 Solution Problem 5 Solution Problem 6 Solution