Negative Binomial Distribution

A Discrete Distribution for an Countably Infinite series of Independent Bernoulli Trials with:

• The same probability of success $p$
• The same probability of failure $q=1-p$ Used for waiting for multiple occurrences of an event.

Random Variable

The RV $X/Y$ counts $\text{\# of failures/trials until }r-th\text{ success}$

PMF

$P_X(x)=\left(\genfrac{}{}{0ex}{}{x+r+1}{r-1}\right)p^r{q}^x,\forall x\in \mathbb{R}$

CDF

There is no CDF

Expectation

$E(X)=\frac{r}{p}$

Variance

${\sigma }^2=\frac{rq}{p^2}$