## Probability Theory: Homework 2

Exercises, due Wed Sept 17.

Jacod and Protter: 4.1, 5.20 and 7.11

4.1 (Poisson Approximation to the Binomial) Let be a Binomial probability with probability of success and the number of trials . Let . Show that

Let and let change so that remains constant. Conclude that for small and large ,

, where .

5.20 Suppose that takes its values in . Show that

.

7.11 Let for a nonnegative function with . Let , a singleton. Show that is a Borel set and also a null set (that is, ).