Binomial Distribution Approximated by Poisson Distribution

Theorem
Let $X$ be a discrete random variable which has the binomial distribution with parameters $n$ and $p$.

Then for $\lambda = n p$, $X$ can be approximated by a Poisson distribution with parameter $\lambda$:


 * $\ds \lim_{n \mathop \to \infty} \binom n k p^k \paren {1 - p}^{n - k} = \frac {\lambda^k} {k!} e^{-\lambda}$

Proof
Let $X$ be as described.

Let $k \in \Z_{\ge 0}$ be fixed.

We write $p = \dfrac \lambda n$ and suppose that $n$ is large.

Then:

Hence the result.