# Thread: Poisson Variables and CLT

1. ## Poisson Variables and CLT

I'm trying to show that for a Poisson random variable $\displaystyle Y$ with mean $\displaystyle \lambda$, we get

$\displaystyle \displaystyle\frac{Y-\lambda}{\sqrt{\lambda}}\Rightarrow N$

as $\displaystyle \lambda\rightarrow\infty$

If $\displaystyle \lambda$ is a natural number n I'm okay, since $\displaystyle Y$ is the sum of n independent Poisson variables with mean 1, and the CLT applies.

My problem is extending to $\displaystyle \lambda$ not necessarily a natural number. Any advice? Thanks!

2. Just let $\displaystyle Y=\sum_{i=1}^{\lambda}X_i$, where I was going to say what you thought.
I assume you wanted $\displaystyle X_i\sim P(1)$.

It still is $\displaystyle {Y-\mu_Y\over \sigma_Y}$ so it should be ok.

Maybe use that same idea .....

Let $\displaystyle Y=\sum_{i=1}^{[\lambda]}X_i +X^*$

$\displaystyle [\lambda]$ is the greatest integer less than or equal to lambda
and $\displaystyle X^*$ is an independent Poisson with mean $\displaystyle \lambda -[\lambda]$
The CLT applies even if the sum doesn't consist of all iid rvs.
And note that

$\displaystyle 0\le {\lambda -[\lambda]\over\lambda}<{1\over\lambda}\to 0$

Just show the CLT applies to $\displaystyle Y=\sum_{i=1}^{[\lambda]}X_i$