## Poisson Distribution

A discrete random variable $X$ is said to have Poisson distribution with parameter $\lambda$ if its probability mass function is $$\begin{equation*} P(X=x)= \left\{ \begin{array}{ll} \frac{e^{-\lambda}\lambda^x}{x!} , & \hbox{x=0,1,2,\cdots; \lambda>0;} \\ 0, & \hbox{Otherwise.} \end{array} \right. \end{equation*}$$

## Mean of Poisson Distribution

The expected value of Poisson random variable is

## Variance of Poisson Distribution

The variance of Poisson random variable is

## Moment Generating Funtion of Poisson Distribution

The moment generating function of Poisson distribution is

## Probability generating function of Poisson Distribution

The probability generating function of Poisson distribution is

## Mode of Poisson distribution

The condition for mode of Poisson distribution is

## Poisson distribution as a limiting form of binomial distribution

In binomial distribution if $n\to \infty$, $p\to 0$ such that $np=\lambda$ (finite) then binomial distribution tends to Poisson distribution