## Poisson approximation to binomial distribution

Let $X$ be a binomially distributed random variable with number of trials $n$ and probability of success $p$.

The mean of $X$ is $\mu=E(X) = np$ and variance of $X$ is $\sigma^2=V(X)=np(1-p)$.

The general rule of thumb to use Poisson approximation to binomial distribution is that the sample size $n$ is sufficiently large and $p$ is sufficiently small such that $\lambda=np$ (finite).

For sufficiently large $n$ and small $p$, $X\sim P(\lambda)$.

The probability mass function of Poisson distribution with parameter $\lambda$ is
```
$$
\begin{equation*}
P(X=x)= \left\{
\begin{array}{ll}
\dfrac{e^{-\lambda}\lambda^x}{x!} , & \hbox{$x=0,1,2,\cdots; \lambda>0$;} \\
0, & \hbox{Otherwise.}
\end{array}
\right.
\end{equation*}
$$
```