Negative Binomial Distribution Calculator
Compute probabilities for waiting times until r successes in repeated Bernoulli trials with success probability p.
P(X = k): 0.104509
P(X ≤ k): 0.684605
P(X ≥ k): 0.315395
Mean: 4.500
Variance: 11.250
Mode: 2
Instructions
- Specify the number of successes you need (r).
- Provide the probability of success in each trial (p).
- Enter k, the failures observed before the r-th success.
- Review point and cumulative probabilities plus distribution moments.
Formula
P(X = k) = C(k + r − 1, r − 1) · pr · (1 − p)k
Mean = r(1 − p) / p
Variance = r(1 − p) / p²
X counts failures before the r-th success. For trials (successes + failures), add r to obtain the total number of trials until the r-th success.
Common Questions
How does this differ from the geometric distribution?
The geometric distribution is a special case with r = 1. The negative binomial extends this to r successes.
What if p varies between trials?
The classical negative binomial assumes constant p. For varying probabilities, other models are needed.
Can r be non-integer?
In probability theory r is typically an integer. Over-dispersion modeling in count data sometimes extends r to positive real numbers (Pascal–gamma mixture), but that requires alternate formulas.