# Poisson and central limit theroem

• Nov 15th 2009, 09:16 PM
cubs3205
Poisson and central limit theroem
Hi, I am having trouble understanding this logic:
If we have n independent Poisson random variables each with a parameter 1/n. Then, the sum of these random variables is a Poisson with parameter 1.
According to the central limit theorem, as n goes to infinity, the distribution of the sum goes to a normal distribution. But in this case, the distribution of the sum still goes to a Poisson with parameter 1, which is not a normal distribution.
Why is there such a contradiction?

Thanks!
• Nov 15th 2009, 11:05 PM
CaptainBlack
Quote:

Originally Posted by cubs3205
Hi, I am having trouble understanding this logic:
If we have n independent Poisson random variables each with a parameter 1/n. Then, the sum of these random variables is a Poisson with parameter 1.
According to the central limit theorem, as n goes to infinity, the distribution of the sum goes to a normal distribution. But in this case, the distribution of the sum still goes to a Poisson with parameter 1, which is not a normal distribution.
Why is there such a contradiction?

Thanks!

If we have n independent Poisson random variables each with a parameter 1/m. Then, the sum of these random variables is a Poisson with parameter n/m. For large (n/m) this is approximatly normal ~N(n/m.n/m), which is fine it agrees with the normal approximation to the Poisson and our expectation from the central limit theorem.

But that was for m fixed. The central limit theorem as usually presented is the limit as the number of iid RV in the sum becomes large (these have fixed distribution), it does not apply when you change the distribution as well as the number of terms in the sum.

(Weasel words are needed if we are going to be rigorous as we can relax some of the conditions in the CLT and still have it work, but you won't usually see such things in a elementary course)

CB