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1. Suppose X has pdf fx(x) = 3x2e-x3 x0. a. Find the cdf of X and show that it is a cdf. b. Find P[X1]. 2. Suppose N and M are independent Poisson random variables with parameters A and 11, respectively, and L=N+M. a. Find the moment generating function, mgf, of N. b. Find the mgf of L. c. Find the distribution of L 3. Suppose the joint density function of the r.v.s X and Y is given by a) Sketch the region in the plane where the joint pdf is positive. b) Determine the marginal distribution of X c) Determine the marginal distribution of Y. d) Sketch the event "X+Y>2" and determine its probability. 4. Let U, V, W be iid Uniform (0, 1), and let M - max{U, V, W ), i.e., M is the largest one of the three. a. Determine the distribution function of M. b. Determine the density function of M. c. Determine E[M]. d. Determine Var(M). 5. Suppose U~Uniform(0,1) and W-Uniform(0,U). a. Determine the conditional distribution of W given U, i.e. Fw1u(w|u) = P[W w|U=u]. b. Determine the density function of W given U. c. Determine the (unconditional) density function of W.

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