## 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

### $E(X) = np$.

## 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

`$M_X(t) = (q+pe^t)^n.$`

## Cumulant Generating Function of Binomial Distribution

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

### $K_X(t) = n\log_e (q+pe^t)$.

## Probability Generating Function of Binomial Distribution

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