Fall 2011 Strategic Practice 11: Section 1 (Law of Large Numbers, Central Limit Theorem) - Question 2
Consider the following:
(a) Explain why the Pois(n) distribution is approximately Normal if n is a large positive integer (specifying what the parameters of the Normal are).
(b) Stirling's formula is an amazingly accurate approximation for factorials: $n!\approx \sqrt{2\pi n}\left ( \frac{n}{e} \right )^{n}$, where in fact the ratio of the two sides goes to 1 as n goes to infinity. Use (a) to give a quick heuristic derivation of Stirling's formula by using a Normal approximation to the probability that a Pois(n) r.v. is n, with the continuity correction: first write P(N = n) = P(n - 1/2 < N < n + 1/2), where N ~ Pois(n).
Solution: Consult iTunes course for full detailed solutions. (a) Use CLT (Central Limit Theorem). (b) Take care to note that the integral is approximately 1 since the interval of integration has length 1 and for large n the integrand is very close to 1 throughout the interval.
"Mathematics is the logic of certainty, but statistics is the logic of uncertainty."