# 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:

 $\ds \lim_{n \mathop \to \infty} \binom n k p^k \paren {1 - p}^{n - k}$ $=$ $\ds \lim_{n \mathop \to \infty} \binom n k \paren {\frac \lambda n}^k \paren {1 - \frac \lambda n}^n \paren {1 - \frac \lambda n}^{-k}$ $\ds$ $=$ $\ds \lim_{n \mathop \to \infty} \frac {n^k} {k!} \paren {\frac \lambda n}^k \paren {1 - \frac \lambda n}^n \paren {1 - \frac \lambda n}^{-k}$ Limit to Infinity of Binomial Coefficient over Power $\ds$ $=$ $\ds \lim_{n \mathop \to \infty} \frac 1 {k!} \lambda^k \paren {1 + \frac {-\lambda} n}^n \paren {1 - \frac \lambda n}^{-k}$ canceling $n^k$ from numerator and denominator $\ds$ $=$ $\ds \dfrac {\lambda^k} {k!} \lim_{n \mathop \to \infty} \paren {1 + \frac {-\lambda} n}^n \paren {1 - \frac \lambda n}^{-k}$ moving $\dfrac {\lambda^k} {k!}$ outside of the limit $\ds$ $=$ $\ds \dfrac {\lambda^k} {k!} \lim_{n \mathop \to \infty} \paren {1 + \frac {-\lambda} n}^n \lim_{n \mathop \to \infty} \paren {1 - \frac \lambda n}^{-k}$ Combination Theorem for Limits of Functions/Product Rule $\ds$ $=$ $\ds \dfrac {\lambda^k} {k!} \lim_{n \mathop \to \infty} \paren {1 + \frac {-\lambda} n}^n \lim_{n \mathop \to \infty} \paren {1 - \frac k n}^{-k}$ $\lambda = n p$ $\ds$ $=$ $\ds \frac {\lambda^k} {k!} e^{-\lambda} \paren {1 }$ Definition of Exponential Function $\ds$ $=$ $\ds \frac {\lambda^k} {k!} e^{-\lambda}$

Hence the result.

$\blacksquare$