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 $$ \begin{equation*} P(X=x)= \left\{ \begin{array}{ll} \dfrac{e^{-\lambda}\lambda^x}{x!} , & \hbox{$x=0,1,2,\cdots; \lambda>0$;} \\ 0, & \hbox{Otherwise.} \end{array} \right. \end{equation*} $$

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