Negative binomial random variable

1. Sablon:Matematika ?

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A negative binomial random variable counts the number of trials needed to get a fixed number of successes in a sequence of independent Bernoulli trials. Its key properties:

• PMF: $$P(X=k)=(\genfrac{}{}{0ex}{}{k+r-1}{r-1})p^r(1-p{)}^k$$ where: - $r$ is the number of successes, - $p$ is the probability of success on each trial, - $k$ is the number of failures before the $r$-th success.

• Mean: $\frac{r(1-p)}{p}$
• Variance: $\frac{r(1-p)}{p^2}$

It generalizes the geometric distribution (when $r=1$). Sablon:Engl