The Central Limit Theorem Date: Sat, 19 Apr 1997 12:10:40 -0400
From: "Quandaries and Queries Consultants"
Sender: Donna

Subject: Probability & Statistics

Help! I am a secondary education mathematics teacher with a probability question:

A skeptic gives the following argument to show that there must be a flaw in the central limit theorem: We know that the sum of independent Poisson random variables follows a Poisson distribution with a parameter that is the sum of the parameters of the summands. In particular, if n independent Poisson random variables, each with parameter 1/n, are summed, the sum has a Poisson distribution with parameter 1. The central limit theoren says the sum tends to a normal distribution, but Poisson distribution with parameter 1 is not normal.

What do you think of this argument?

Donna

Thanks for your input.

Hi Donna

The Central Limit Theorem says that for any given distribution (with finite variance), there exists an N such that the sum of N independent copies of random variables from this distribution is "approximately normal". It is important that the size of N depends on the particular distribution. For a Poisson distribution with parameter 1, you get approximate normality with N=30, but for a Poisson distribution with parameter 1/100, you would need N=3000. In your example for a Poisson distribution with parameter 1/n you get approximate normality with N=30n.