# Negative Binomial Distribution

A discrete probability distribution $$NegBin(r, p)$$ having the PMF
$p_{r,p}(k) = {}^{k+r-1}C_k \times p^r \times (1-p)^k$
that governs the probability that, for independent experiments with a probability of success $$p$$, there are $$k$$ failures before there are $$r$$ successes, where $$k$$ is greater than or equal to zero, $$r$$ is greater than zero and $${}^nC_k$$ is the combination of $$k$$ from $$n$$ objects. If $$r$$ equals one then it is identical to the geometric distribution with a probability of success equal to $$p$$.
It can be generalised to non-integer $$r$$ with
$p_{r,p}(k) = \frac{\Gamma(k+r)}{k! \times \Gamma(r)} \times p^r \times (1-p)^k$
where the exclamation mark stands for the factorial and $$\Gamma$$ is the gamma function

### Gallimaufry  AKCalc ECMA  Endarkenment Turning Sixteen

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