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

$P_X(t)=p(1-qt)^{-1}$.

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