Why the Poisson and Binomial distributions are approximately normal?

Question

Why the Poisson distribution P(n) and Binomial distribution B(n, p) are approximately normal if n is a large positive integer and p ∈ (0, 1) is fixed? I found the examples of normal approximations, but didn't find the explanation why does it work and what will be the parameters of these approximations respectively.

Answer 1

1. Let's start with the question on Binomial.

As you probabily know, the binomial distribution can be expressed as the sum of $n$ independent Bernoulli. Then you can apply the central limit theorem in the version of De Moivre Laplace

2. As per the Poisson is concerned, it can be wiewed a limit Law of the binomial thus what explained in 1. is still valid (when $n$ is great enough)

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Both statement in 1. and 2. can be easily proved with MGF's properties

Proof of 2.

Setting $p=\frac{\theta}{n}$ the MGF of a Binomial $B(n;p)$ is the following

$$\Big(1-\frac{\theta}{n}+\frac{\theta}{n}e^t\Big)^n=\Big[1+\frac{\theta(e^t-1)}{n}\Big]^n$$

It is evident that

$$\lim\limits_{n \to \infty}\Big[1+\frac{\theta(e^t-1)}{n}\Big]^n=e^{\theta(e^t-1)}$$

Which is exactly the MGF of a Poisson distribution.

Proof of 1.

simply multiply n times MGF of the $B(1;p)$ and see that the result is the MGF of the $B(n;p)$