$P(X = k) = \binom{k + r - 1}{k}(1-p)^r p^k, \quad k = 0, 1, 2, \ldots$ $\text{where } r = \text{number of successes (target)},\quad p = \text{probability of failure per trial}$ The Negative Binomial Distribution $X \sim \text{NegBin}(r, p)$ models the number of failures before the $r$-th success in a sequence of independent Bernoulli trials, each with success probability $1-p$. Its mean is $E[X] = rp/(1-p)$ and variance is $\text{Var}(X) = rp/(1-p)^2$. Because $\text{Var}(X) > E[X]$, the negative binomial exhibits **overdispersion** relative to the Poisson and is widely used in actuarial claim count models. > [!example]- Number of Non-Claims Before Third Claim {💡 Example} > An insurer processes files where each independently results in a claim with probability $1 - p = 0.4$. What is the probability that exactly 2 non-claims occur before the 3rd claim? > > > [!answer]- Answer > > Here $r = 3$, $p = 0.6$ (probability of non-claim), $k = 2$: > > $P(X = 2) = \binom{2+3-1}{2}(0.6)^3(0.4)^2 = \binom{4}{2}(0.216)(0.16) = 6 \times 0.03456 \approx 0.2074$