central limit theorem

Question

Let X1, , X2 , X3 , …, X20 , be independent samples of a Poisson RV with mean 1. Use the
central limit theorem to approximate

P(∑_(i=1) to 20?Xi≥15)?

Answer 1

Each X_i follows a Poisson distribution of mean µ=1 and variance σ²=1.

By the central limit theorem, the distribution Y= Σ_i=1ⁿ X_i as n gets large approaches a normal distribution of mean nµ and variance nσ². Our n=20 is not large, so Y=Σ_i=1²⁰ X_i only follows approximately a normal distribution of mean 20 and variance 20. Find the z-score for Y=15: z = (15-20)/√(20) = -1.118, then

P(Y≥15) = P(z≥-1.118) = 0.868

 

This is the approximate probability the problem asked for, however, we can also give the exact answer, since it is well-known that the sum Y of n independent Poisson distributions of mean µ_i follows a Poisson distribution of mean ∑µ_i, so in our case Poi(20). You can use the poissoncdf(20,15) function on your calculator to find the complement probability that Y≤15, so that the exact answer is

P(Y≥15) =1 - poissoncdf (20,15) = .84348.