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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