What is the negative binomial distribution?

It is a discrete probability distribution that models the number of trials needed to achieve r successes, where each trial independently succeeds with probability p. The PMF gives the probability that the r-th success falls exactly on trial k.

What is the difference between PMF and CDF?

The PMF (P(X=k)) gives the probability of the r-th success occurring on exactly trial k. The CDF (P(X<=k)) gives the probability it occurs on or before trial k, summing all PMF values from r to k.

Why must k be at least r?

You need at least r trials to achieve r successes, so k < r is impossible and the probability is 0.

How is this related to the geometric distribution?

The geometric distribution is a special case of the negative binomial with r=1, modeling the number of trials needed for the first success.

What are the mean and variance formulas?

Mean = r / p. Variance = r * (1 - p) / p^2. Both increase as p decreases (harder to succeed) or r increases (more successes needed).

Negative Binomial Distribution Calculator

Name: Negative Binomial Distribution Calculator
Availability: InStock
Author: Nham Vu

Enter success count r, trial count k, and success probability p to get the PMF, CDF, mean, and variance of the negative binomial distribution.

Parameters

r — Number of successes

Integer ≥ 1

k — Trial of the r-th success

Integer ≥ r

p — Probability of success

Between 0 and 1 (inclusive)

Enter values and click Calculate to see results.

Summary

Enter success count r, trial count k, and success probability p to get the PMF, CDF, mean, and variance of the negative binomial distribution.

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

Related tools

Binomial Distribution Poisson Distribution Geometric Distribution

Last updated: 2026-05-23 · Reviewed by Nham Vu