X is Binomial(1000, 0.001). Compute P(X >= 5) and compare two approximations, the normal with a continuity correction and the Poisson. Which one fails and why?

X is Binomial(1000, 0.001). Compute P(X >= 5) and compare two approximations, the normal with a continuity correction and the Poisson. Which one fails and why?

Approach: Compute the exact tail, then check the shape assumptions each approximation makes by computing the skewness of this binomial and asking how many values carry the bulk of the mass.

The exact value is 0.00364, the Poisson approximation gives 0.00366, and the normal approximation gives 0.00023, understating the tail by a factor of 16. With n p = 1 and standard deviation 0.9995, the normal approximation with continuity correction asks for P(Z > (4.5 - 1)/0.9995) = P(Z > 3.5) = 2.3 * 10^{-4}. The binomial has skewness (1 - 2p)/sqrt(n p (1-p)) = 1.0, far from the symmetric shape the normal assumes, and the count is a non-negative integer with mean 1, so four values hold almost all the mass. The Poisson limit applies because n is large and p is small with n p fixed, giving 1 - e^{-1}(1 + 1 + 1/2 + 1/6 + 1/24) = 0.00366. Normal tails become usable only once n p (1-p) reaches about 10.

Follow-up: With n p held at 1, does the normal approximation ever become accurate as n grows?

Key concepts: normal approximation, poisson limit, skewness, continuity correction.