Binomial Distribution Approximated by Poisson Distribution

Theorem

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

Suppose $n$ is "very large" and $p$ is "very small", but $np$ of a "reasonable size".

Then $X$ can be approximated by a Poisson distribution with parameter $\lambda$ where $\lambda = np$.

Proof

Let $X$ be as described.

Let $k \ge 0$ be fixed.

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

Then:

 $\ds \map \Pr {X = k}$ $=$ $\ds \binom n k p^k \paren {1 - p}^{n - k}$ $\ds$ $\simeq$ $\ds \frac {n^k} {k!} \paren {\frac \lambda n}^k \paren {1 - \frac \lambda n}^n \paren {1 - \frac \lambda n}^{-k}$ When $n >> k$, it is a reasonable approximation for $\dbinom n k$ $\ds$ $=$ $\ds \frac 1 {k!} \lambda^k \paren {1 + \frac {-\lambda} n}^n \paren {1 - \frac \lambda n}^{-k}$ $\ds$ $=$ $\ds \frac 1 {k!} \lambda^k \paren {1 + \frac {-\lambda} n}^n$ as $1 - p = \paren {1 - \dfrac \lambda n}$ is very close to $1$ $\ds$ $\simeq$ $\ds \frac 1 {k!} \lambda^k e^{-\lambda}$ Definition of Exponential Function

Hence the result.

$\blacksquare$

Comment

Okay wise guy, exactly what constitutes "very large", "very small", and "of a reasonable size"?

Well, if $n = 10^6$ and $p = 10^{-5}$, we have $np = 10$.

That's the sort of order of magnitude we're talking about here.