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