Negative Binomial Distribution
A discrete random variable $X$ is said to have negative binomial distribution if its p.m.f. is given by
$$ \begin{aligned} P(X=x)&=\binom{x+r-1}{r-1} p^{r} q^{x},\\ & \qquad x=0,1,2,\ldots; r=1,2,\ldots\\ & \qquad\qquad 0<p, q<1, p+q=1. \end{aligned} $$
Mean of Negative Binomial Distribution
The mean of negative binomial distribution is
$E(X)=\dfrac{rq}{p}$.
Variance of Negative Binomial Distribution
The variance of negative binomial distribution is
$V(X)=\dfrac{rq}{p^2}$.
For negative binomial distribution $V(X)> E(X)$, i.e., variance > mean.
MGF of Negative Binomial Distribution
The MGF of negative binomial distribution is
$M_X(t)=\big(Q-Pe^{t}\big)^{-r}$.
CGF of Negative Binomial Distribution
The CGF of negative binomial distribution is
$K_X(t)=-r\log_e(Q-Pe^t)$.
Characteristics Function of negative binomial distribution
The characteristics function of negative binomial distribution is
