Negative binomial distribution

Flip a biased coin with success probability $p$ until you accumulate exactly $r$ successes. The number of failures along the way is negative-binomial distributed.

P (K = k) = (k k + r - 1) (1 - p)^{k} p^{r}

r (successes to wait for) 3 p (success prob) 0.40 n (samples) 1000

Dark bars: PMF

P (K = k) = (k k + r - 1) (1 - p)^{k} p^{r}

. Light bars: empirical frequencies from 1000 simulated waits. Mean

E [K] = r (1 - p) / p

= 4.50.

What to notice

Raising $r$ shifts mass to the right (you’re waiting for more successes, so more failures pile up) and smooths the shape toward a Normal, since the total is a sum of $r$ independent geometric waits.
Lowering $p$ stretches the whole distribution — rare successes mean long waits.
Variance exceeds the mean by a factor of $1/ p$ . Compared to Poisson (where variance equals the mean), the negative binomial is overdispersed — which is why it’s the go-to for count data that clumps.

Relationship to other distributions

Geometric. With $r = 1$ , you’re waiting for the first success, which is exactly the geometric distribution on non-negative integers.
Sum of geometrics. More generally, a negative binomial with parameter $r$ is the sum of $r$ independent geometrics — a discrete cousin of the gamma distribution, which is the sum of exponentials.
Poisson-gamma mixture. A Poisson whose rate is itself gamma-distributed integrates out to a negative binomial. That’s the Bayesian reason it’s used whenever Poisson rates vary between observations.

E [K] = r (1 - p) / p

Var (K) = r (1 - p) / p^{2}