## Geometric Distribution

A discrete random variable $X$ is said to have geometric distribution if its probability mass function is given by $$\begin{equation*} P(X=x) =\left\{ \begin{array}{ll} q^x p, & \hbox{x=0,1,2,\ldots} \\ & \hbox{0<p,q<1, p+q=1} \\ 0, & \hbox{Otherwise.} \end{array} \right. \end{equation*}$$

## Mean of Geometric Distribution

The mean of geometric distribution is

## Variance of Geometric Distribution

The variance of geometric distribution is

### $V(X) =\dfrac{q}{p^2}$.

For geometric distribution, variance > mean.

## MGF of Geometric Distribution

The moment generating function of geometric distribution is

## Cumulant Generating Function

The cumulant generating function of geometric distribution is

## Characteristics function of Geometric Distribution

The characteristics function of geometric distribution is

## Probability generating function of Geometric Distribution

The probability generating function of geometric distribution is