Negative binomial definition and characteristics

Definition

In a sequence of independent Bernoulli trials with a constant probability of success p (and q = 1 − p), the random variable X, the number of trials needed to obtain a fixed number r of successes, follows the negative binomial distribution:

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

Reasoning: the x-th trial is the r-th success (probability p), and in the first x − 1 trials exactly r − 1 are successes, in C(x − 1, r − 1) ways with probability pʳ⁻¹ qˣ⁻ʳ.

Important characteristics

1. Parameters: r (number of successes wanted) and p (probability of success).
2. Conditions: independent trials, two outcomes, constant p, trials continue until r successes occur.
3. Range: X takes the values r, r + 1, r + 2, … (discrete and unbounded).
4. Mean: E(X) = r/p.
5. Variance: Var(X) = rq/p², so the variance is larger than the mean when q > p. The standard deviation is √(rq)/p.
6. Special case: when r = 1 it becomes the geometric distribution, P(X = x) = q ˣ⁻¹ p, with mean 1/p.
7. Additive property: the sum of independent NB(r₁, p) and NB(r₂, p) variables is NB(r₁ + r₂, p).
8. Alternative form: if Y = X − r (the number of failures before the r-th success), P(Y = y) = C(y + r − 1, y) pʳ qʸ with mean rq/p. The name "negative binomial" comes from the fact that the probabilities are the terms of the expansion of pʳ (1 − q)⁻ʳ.
9. The probabilities add to 1: Σ C(x − 1, r − 1) pʳ qˣ⁻ʳ = 1.

Example: the probability that a coin gives the 2nd head on the 4th toss (p = 0.5) is C(3, 1)(0.5)²(0.5)² = 3/16 = 0.1875. The expected number of tosses to get 2 heads is 2/0.5 = 4.