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 limiting distribution of the Poisson$(\lambda)$ distribution as $\lambda \rightarrow \infty$ is normal.
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$(0,1)$.