Discrete Negative Binomial Distribution

A kind of Discrete Distribution that models the number of independent Bernoulli Trial until the $r^{\text{th}}$ Success, where each trial has a Probability $p$ of success.

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

• $f(x)=\left(\genfrac{}{}{0ex}{}{x-1}{r-1}\right)p^r{q}^{x-r}\text{for }x\in {\mathbb{N}}_0+r$
• $E(X)=\frac{r}{p}$
• $\text{Var}(X)=\frac{r}{p}\left(\frac{1}{p}-1\right)=\frac{r(1-p)}{p^2}$
• $X_{\text{NB}}={\sum }_{i=1}^r{X}_i\sim \text{Geo(p)}$