Homework 3 (due on Fr 11 June 2021) Please, hand in your solutions via Nestor until Fr 11 June, 11.00 AM. If you hand in late then that will count as not handed in. Problem 1; 2 points. 1. Let T ~ Exp() ) be an exponential random variable with parameter > > 0. Show that the mof of T is given by Mr(t) = (1-1/2)-1. 2. Let Z ~ / (0,a) be a normal random with variance a > 0. Show that the mef of Z- is given by Mzz(t) = (1-2at) -1/2. Problem 2; 3 points. For n > 1 let X,, be a Poi(n) random variable. 1. Show that for every k > 1 there exist iid random variables Yin, ..., Yan such that Yin + ... + Yk,n has the same distribution as X,. 2. Let & E (0, 1) be arbitrary. Compute lim, .. P(n(1 - 6) 1 different brands. More precisely, you have one bar from brand number 1, two bars from brand number 2, ..., and m bars from brand number m. You pick a bar at random and let X, E {1,...,m} be the brand number of the chosen bar. 1. Compute the pmf and mgf of Xm/m. 2. Compute the expectation and variance of Xm/m. 3. Show that Xm/m converges in distribution as m -co and compute the cdf of the limiting distribution. Problem 4; 2 points. Little Bart sends a wishlist to Sinterklaas consisting of 9 items. Sinterklaas will satisfy a random subset of these wishes. More precisely, Sinterklaas chooses a subset of a random size X E {0,...,9} with probability fx (k) = c(x) k for some c > 0. 1. Determine c. 2. Compute the mgf of X