Question 1

What does the negative binomial distribution model?

Accepted Answer

The negative binomial distribution answers: "How many failures occur before achieving exactly r successes?" Each trial is independent with constant success probability p. Examples: (1) How many coin flips fail before getting 3 heads? (2) How many defective items are inspected before finding 5 good ones? (3) In A/B testing, how many users see the control variant before r conversions in the test variant? (4) Count data with overdispersion (more variance than Poisson).

Question 2

What is the difference between negative binomial and geometric distributions?

Accepted Answer

The geometric distribution is the special case of the negative binomial with r = 1: it models the number of failures before the first success. The negative binomial with r > 1 models failures before the r-th success. The geometric distribution has PMF P(X = k) = (1 − p)^k × p and mean (1 − p)/p. The negative binomial generalises this by requiring r successes instead of just 1.

Question 3

What are the mean and variance of the negative binomial distribution?

Accepted Answer

Mean = r(1 − p) / p. Variance = r(1 − p) / p². Note: Variance / Mean = 1/p > 1. This is why the negative binomial is used for overdispersed count data — it always has variance greater than mean, unlike the Poisson (where variance = mean). As p → 1, variance → 0. As p → 0, variance → ∞ for fixed mean.

Question 4

Why is it called "negative binomial"?

Accepted Answer

The name comes from the negative binomial series expansion: (1 − q)^{−r} = Σ C(k+r−1, k) q^k. The PMF P(X = k) = C(k+r−1, k) p^r (1−p)^k uses coefficients from this expansion. Alternatively, think of it as the binomial distribution but "negated" — instead of fixing the number of trials and counting successes, you fix the number of successes and count trials (or failures).

Question 5

When does the negative binomial approximate Poisson?

Accepted Answer

As r → ∞ and p → 1 with r(1 − p) = λ fixed, the negative binomial converges to Poisson(λ). In practice, the negative binomial is used when count data shows overdispersion (variance > mean) that the Poisson cannot model. In ecology, genetics, and epidemiology, the negative binomial is the standard alternative to Poisson regression when counts are overdispersed.

Question 6

How do I compute P(X ≤ k) for the negative binomial?

Accepted Answer

The CDF P(X ≤ k) = Σ P(X = j) for j = 0 to k. This sum has no simple closed form and must be computed directly. This calculator uses exact summation. Alternatively, the CDF equals the regularized incomplete beta function: I_p(r, k + 1), where I is the regularized incomplete beta. This relationship connects to the binomial CDF: P(X ≤ k) = P(Binomial(k+r, p) ≥ r).

Distribution	Models	Variance vs Mean	Use when
Binomial	k successes in n trials	Var < Mean (unless p = 0.5)	Fixed n, independent trials
Geometric	Failures before 1st success	Var = (1−p)/p² > Mean	First-success timing
Negative Binomial	Failures before r-th success	Var > Mean (overdispersion)	Overdispersed count data
Poisson	Events in time/space	Var = Mean (equidispersion)	Rare events, fixed rate

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