MATH 367, fall 98, CHAPTER 6 Section 1: Poisson Approximation Section 2: Stirling’s Formula

Problem 1: A Country Club has 4000 members, 2000 men and 2000 women. Exactly half of the members will be randomly chosen to receive a free ticket to a Boston Pop’s performance. All members have the same chance of getting a ticket. What is the probability the exactly 1000 men and 1000 women get tickets? Definition: We say that f(n) ∼ g(n) that is, g approximates f assymptotically, if

lim_{n→∞} \frac{f(n)}{g(n)} = 1.

Poisson’s Approximation:

\binom{n}{k}p^{k}q^{n−k} ⟶ e^{−m} \frac{m^{k}}{k!} as n → ∞,

where p = m/n. This means that if X is B(n, p), where n is large then and p = m/n, then P (X = k) ∼ e^{−m} \frac{m^{k}}{k!} .

2. Use the Poisson approximation to estimate the answer to Problem 1.

Definitions 0.1. X is a Poisson random variable with parameter m if its distribution mass function is P (X = k) = e^{−m}m^{k}/k! for k = 0, 1, 2, 3, . . ..

Note: If X is Poisson with parameter m, E(X) = m.

Stirling’s Formula: n! ∼ n^{n}e^{−n}\sqrt{2πn}.

3. Use the Stirling’s formula to estimate the answer to Problem 1.

Homework Section 6.1: read section 6.1 and do problems 1–7, 9, 10. Homework Section 6.2: read section 6.2 and do problems 16, 18–21.