## Binomial Distribution

A discrete random variable $X$ is said to have Binomial distribution with parameter $n$ and $p$ if its probability mass function is $$\begin{equation*} P(X=x)= \left\{ \begin{array}{ll} \binom{n}{x} p^x q^{n-x}, & \hbox{x=0,1,2,\cdots, n;} \\ & \hbox{0<p<1, q=1-p} \\ 0, & \hbox{Otherwise.} \end{array} \right. \end{equation*}$$

## Mean of Binomial Distribution

The mean or expected value of binomial random variable $X$ is

## Variance of Binomial distribution

The variance of Binomial random variable $X$ is

### $V(X) = npq$.

For Binomial distribution Mean > Variance.

## Moment Generating Function of Binomial Distribution

The moment generating function (MGF) of Binomial distribution is given by

## Cumulant Generating Function of Binomial Distribution

The cumulant generating function of Binomial random variable $X$ is

## Probability Generating Function of Binomial Distribution

The probability generating function (PGF) of Binomial distribution is given by