Exercises for Wed 3 September

1. On Wed 20 August I gave a general introduction to the course, indicating themes to come, etc. Again, the curriculum will largely consist of Chs 3, 5, 6, 7, 8, 9, 10, with Chs 1, 2, 4 defined as "cursory curriculum". I spent some time discussing material from Chs 1-2, which should be seen as giving the essential background from probability theory, with random variables, distributions, independence and dependence, expectation operators, transformations, etc. 

2. Next week I'm at a conference in Wien, and there will be no teaching for stk 4011/9011 then. Spend time reading Chs 3 and 5, and work with these exercises, which will be discussed Wed 3 September:

Ch 1: 6, 20, 25, 47, 53, 55. Ch 2: 1, 6, 12, 22, 28.

Extra: (a) Let X be a Poisson with parameter lambda. Find the moment-generating function M(t). (b) Let Y_n be binomial(n, p) with np tending to lambda. Show that the mgf of Y_n tends to M(t). (c) Let Z_n = X_1 + ... + X_n, with these being independent and X_i being bin(1, p_i). Assume that \sum p_i tends to \lambda and that \max p_i tends to zero. Show that the mgf of Z_n tends to M(t).