## 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

### $E(X)=\lambda$.

## Variance of Poisson Distribution

The variance of Poisson random variable is

### $V(X) =\lambda$.

## Moment Generating Funtion of Poisson Distribution

The moment generating function of Poisson distribution is

### $M_X(t)=e^{\lambda(e^t-1)}$.

## Probability generating function of Poisson Distribution

The probability generating function of Poisson distribution is

### $P_X(t)=e^{\lambda(t-1)}$.

## Mode of Poisson distribution

The condition for mode of Poisson distribution is

### $\lambda-1 \leq x\leq \lambda$.

## 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