High Lambda Poisson approaches Normal Distribution

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In my Generalized Linear Model class, we were tasked with proving that with high $\lambda$, the Poisson distribution approaches the standardized normal distribution. I went ahead and wrote up the proof for the extra credit and really enjoyed doing it. I’ll note that converting the exponential to a series was not my idea and had a little Stack Exchange inspiration. It reminded me how useful series expansions were, so that will hopefully prove useful in future research.

The Theorem

The limiting distribution of the Poisson$(\lambda)$ distribution as $\lambda \rightarrow \infty$ is normal.

The Proof

Let $X \sim Poisson(\lambda)$ which has the probability mass function

and moment generating function

We will specifically consider the standardized Poisson random variable $X$

which has the Moment Generating Function

Now, we take the limit as $\lambda$ approaches $\infty$ and utilize the taylor series expansion

Therefore, we have

To find the limiting moment generating function, we take the limit of this moment generating function as $\lambda \rightarrow \infty$ which results in

which is the moment generating function of N$(0,1)$.