## 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