Negative Binomial Distribution

Definition

$X\sim \text{NB}(r,p)$

Distribution of number of failures ($x$) needed to get the $r$-th success

• pmf: $\left(\genfrac{}{}{0ex}{}{x+r-1}{r-1}\right)p^r(1-p{)}^x$, $x\in 0,1,2,\dots$
• mean: $\frac{r(1-p)}{p}$
• var: $\frac{r(1-p)}{p^2}$
• mgf: $p^r[1-(1-p)e^t{]}^{-r}$, $t<-\ln (1-p)$