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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 P be a Binomial probability with probability of success p and the number of trials n. Let \lambda = pn. Show that

\displaystyle P(k \text{ successes}) = \frac{\lambda^k}{k!} \left(1 - \frac{\lambda}{n}\right)^n \left[\left(\frac{n}{n}\right) \left(\frac{n-1}{n} \right) \cdots \left(\frac{n - k + 1}{n}\right) \right] \left(1 - \frac{\lambda}{n}\right)^{-k}.

Let n \to \infty and let p change so that \lambda remains constant. Conclude that for small p and large n,

\displaystyle P(k \text{ successes}) \approx \frac{\lambda^k}{k!} e^{-\lambda}, where \lambda = pn.

5.20 Suppose that X takes its values in \mathbb{N} = \{0, 1, 2, \ldots \}. Show that

\displaystyle E[X] = \sum_{n=0}^\infty P(X > n).

7.11 Let \displaystyle P(A) = \int_{-\infty}^\infty 1_A(x) f(x) dx for a nonnegative function f with \displaystyle \int_{-\infty}^\infty f(x) dx = 1. Let A = \{x_0\}, a singleton. Show that A is a Borel set and also a null set (that is, P(A) = 0).

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