Negative binomial: definition and difference

Definition

Consider repeated independent Bernoulli trials, each with probability of success p and failure q = 1 − p. The negative binomial distribution gives the probability that the r-th success occurs on the x-th trial.

P(X = x) = C(x − 1, r − 1) pʳ qˣ⁻ʳ ,   x = r, r + 1, r + 2, ...

The last (x-th) trial must be a success, and the previous x − 1 trials contain exactly r − 1 successes. Here r is a fixed positive integer and p is the parameter. It is written NB(r, p).

Characteristics

1. The trials are independent and have only two outcomes, with constant p.
2. The number of successes r is fixed; the number of trials X is random.
3. The range of X is r, r + 1, r + 2, … (infinite and discrete).
4. Mean: E(X) = r/p.
5. Variance: Var(X) = rq/p², which is greater than the mean when q > p.
6. For r = 1 it reduces to the geometric distribution: P(X = x) = q ˣ⁻¹ p.
7. The sum of r independent geometric variables is negative binomial.
8. If the number of failures Y = X − r is counted, P(Y = y) = C(y + r − 1, y) pʳ qʸ, with mean rq/p and variance rq/p².

Example: a player shoots until he scores 3 goals, with p = 0.4 per shot. The probability that the 3rd goal comes on the 5th shot is C(4, 2) (0.4)³ (0.6)² = 6 × 0.064 × 0.36 = 0.138.

Difference from the binomial distribution