Negative Binomial Distribution Calculator
Enter success count r, trial count k, and success probability p to get the PMF, CDF, mean, and variance of the negative binomial distribution.
Use the Negative Binomial Distribution Calculator
Parameters
Integer ≥ 1
Integer ≥ r
Between 0 and 1 (inclusive)
PMF P(X=k)
—
CDF P(X≤k)
—
Mean
—
Variance
—
Formula Breakdown
PMF Distribution (k from r to r+30)
Enter values and click Calculate to see results.
Summary
The negative binomial distribution models the number of trials needed to achieve a fixed number of successes in a series of independent Bernoulli trials. This calculator computes the probability mass function (PMF) — the exact probability that the r-th success occurs on the k-th trial — as well as the cumulative distribution function (CDF), expected mean, and variance. It is widely used in statistics, quality control, epidemiology, and reliability engineering.
How it works
- Enter r (number of successes required), k (trial number of the r-th success), and p (probability of success per trial).
- The calculator validates that k >= r and 0 < p <= 1.
- It computes the PMF: P(X=k) = C(k-1, r-1) * p^r * (1-p)^(k-r).
- The CDF P(X <= k) sums all PMF values from k=r up to the entered k.
- Mean (r/p) and variance (r*(1-p)/p^2) are displayed alongside the PMF chart.
Use cases
- Determine the probability that the 5th defective item is found on the 20th inspection.
- Model the number of sales calls needed to close a target number of deals.
- Estimate how many trials are needed before achieving a certain number of successes in a clinical trial.
- Analyze failure patterns in reliability engineering and quality assurance.
- Model overdispersed count data in epidemiology and ecology.
- Compute waiting-time probabilities in queuing and scheduling models.
Frequently Asked Questions
Last updated: 2026-05-23 · Reviewed by Nham Vu