## 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

### $E(X)=\dfrac{q}{p}$.

## 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

### $M_X(t) = p(1-qe^t)^{-1}$.

## Cumulant Generating Function

The cumulant generating function of geometric distribution is

### $K_X(t)=\log_e \bigg( \frac{p}{1-qe^t}\bigg)$.

## Characteristics function of Geometric Distribution

The characteristics function of geometric distribution is

### $\phi_X(t)=p(1-qe^{it})^{-1}$.

## Probability generating function of Geometric Distribution

The probability generating function of geometric distribution is