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Negative Binomial

r2r \geq 2

X# of trials until the rth successX - \text{\# of trials until the rth success}

xZ+x \in \mathbb{Z}^+

x{r, r+1, r+2,}x \in \{r, ~r+1, ~r+2, \cdots\}

P(X=x)=(x1r1)pr1(1p)xrpP(X=x) = {{x-1} \choose {r-1}} p^{r-1} (1-p)^{x-r} p

Where (x1r1)pr1(1p)xr{{x-1} \choose {r-1}} p^{r-1} (1-p)^{x-r} is the r1r-1 successes out of x-1 trials, and pp is the xxth trial of rrth success.