Poisson Distribution Homework: Use Central Limit Theorem

[rhyno89]

Homework Statement

We can approximate a poisson distribution from the normal. Suppose lambda is a large positive value; let X ~ Poisson(lambda) and let X1...Xn be independant identicly distributed from a Poisson (lambda/n) distribution. Then X and X1+...+Xn have the same distribution. Use the central limit theorem to determine the approximate distribution of X...

Homework Equations

The Attempt at a Solution

I know that the above statement is true through moment generating functions. If X is distributed with mean of lambda then n random variables with a mean of lambda over n will sum to a distribution with mean of lambda.

It seems from the question that the distribution of X would just be a poisson distribution with mean of lambda but that doesn't really require any work as the problem states that.

Any help on getting started?

 

[mighty2000]

I can confirm that the statement is true. This is because as the sample size (n) increases, the distribution of the sum of n independent and identically distributed random variables approaches a normal distribution, as stated by the Central Limit Theorem. In this case, the sum of n Poisson distributed random variables with mean lambda/n will approach a normal distribution with mean n(lambda/n) = lambda and variance n(lambda/n) = lambda. This is the same distribution as X, which is also Poisson with mean lambda. Therefore, we can approximate a Poisson distribution from the normal distribution.