Negative Binomial Distribution Calculator - PMF, CDF, Mean, and Variance

A negative binomial distribution calculator turns the target success count r, the per-trial success probability p, and the observed failure count k into the exact probability, cumulative probability, survival, and the full set of summary statistics for the discrete distribution of failures before the r-th success in a sequence of independent Bernoulli trials. Use it for quality control lots, sports win streaks, insurance claim counts, and overdispersed count data, then read the PMF, CDF, mean, variance, mode, and skewness from the same three inputs.

• • Quality control inspection: Estimate how many defective items to expect before a target number of good items in a sample, when each draw is independent with constant defect probability.
• • Sports analytics: Model the number of losses a team or player accumulates before reaching a target number of wins in a season of independent games with a stable win rate.
• • Epidemiology and insurance claims: Fit overdispersed count data such as doctor visits, accidents, or claims per period, where the variance is larger than the mean.
• • Variance always exceeds the mean: Because σ² = μ/p with p < 1, the negative binomial distribution is always overdispersed relative to the Poisson distribution with the same mean.

Because the negative binomial distribution fixes the number of successes and counts the failures, the Binomial Distribution Calculator is the natural pair to revisit when the experiment instead fixes the number of trials and counts the successes.

The calculator reads r, p, and k, builds the exact PMF from the binomial coefficient C(r + k − 1, k) and the powers p^r and (1 − p)^k, then sums the PMF from 0 up to k to produce the cumulative probability and the survival. The summary measures all come from closed-form expressions in r and p alone.

• r: Target number of successes. Positive integer that controls how many successes must be observed before the count stops.
• p: Success probability on each independent Bernoulli trial. Strictly between 0 and 1.
• k: Number of failures observed before the r-th success. Non-negative integer.
• C(r + k − 1, k): Binomial coefficient in the PMF, equal to the number of ways to interleave k failures with r − 1 successes in the first r + k − 1 trials.
• p^r: Probability that the last r trials all succeed and contain the r-th success.
• (1 − p)^k: Probability that exactly k failures occur among the first r + k − 1 trials.

The binomial coefficient is computed with a multiplicative product from i = 1 through k so the calculation stays accurate for k as large as 10 000 without overflowing.

According to Wolfram MathWorld's Negative Binomial Distribution entry, the distribution (also called the Pascal or Pólya distribution) gives the probability of r − 1 successes and k failures in r + k − 1 trials with success on the (r + k)th trial, with PMF C(r + k − 1, k) p^r (1 − p)^k, mean r(1 − p)/p, variance r(1 − p)/p², skewness (2 − p)/√(r(1 − p)), and collapses to the geometric distribution when r = 1.

When r = 1 the negative binomial distribution collapses to the geometric distribution, so the Geometric Distribution Calculator is the right sanity check for any single-success waiting-time problem you model here.

Four ideas separate a negative binomial calculation from a black-box probability and let the user interpret the result on real data.

These four ideas are also why the negative binomial distribution is the right tool for overdispersed count data that the Poisson distribution cannot fit cleanly.

Because the negative binomial distribution is overdispersed relative to a Poisson count, the Poisson Distribution Calculator is the natural baseline to compare against whenever you want to see how much extra spread the negative binomial adds.

Enter the target successes r, the success probability p, and the failure count k to read the exact probability, cumulative probability, and summary measures in one panel.

1. 1 Set the target successes r: Type a positive integer for the number of successes you need before the count stops. r = 1 reduces the distribution to the geometric distribution.
2. 2 Set the success probability p: Type the per-trial success probability strictly between 0 and 1. The calculator rejects p ≤ 0 or p ≥ 1 with a clear validation error.
3. 3 Enter the failure count k: Type the observed number of failures before the r-th success. k = 0 means the first r trials were all successes.
4. 4 Read the exact and cumulative probabilities: The PMF, CDF, survival, and the two strict-comparison probabilities update from r, p, and k automatically.
5. 5 Read the summary measures: The mean, variance, standard deviation, mode, and skewness update from r and p alone, so changing only k does not affect them.
6. 6 Reset to defaults: Press reset to return r, p, and k to 3, 0.5, and 2.

Suppose r = 3, p = 0.5, and k = 2. The PMF is 0.1875, the CDF is 0.5, the mean is 3, the variance is 6, and the mode is 2, with the variance being twice the mean as the visual fingerprint of an overdispersed count.

When you want to compare the observed failure count with the distribution's mean and standard deviation, the Z-Score Calculator returns the standardized score that lets you judge how unusual the observation is.

An all-in-one negative binomial distribution calculator removes the need to switch between separate PMF, CDF, mean, and variance tools.

• • Exact and cumulative probabilities from the same inputs: Read P(X = k), P(X ≤ k), P(X ≥ k), P(X > k), and P(X < k) from one set of r, p, and k inputs without re-typing values.
• • Closed-form summary measures: Read the mean, variance, standard deviation, mode, and skewness from r and p alone, with no additional assumptions.
• • Auditable binomial coefficient: Display C(r + k − 1, k) alongside the PMF so the user can verify the combinatorial count by hand.
• • Geometric reduction for r = 1: Setting r = 1 turns the calculator into a geometric distribution calculator with the same closed-form formulas.
• • Strict input validation: Out-of-range r, p, and k are rejected with clear error messages so silent overflow or degenerate distributions do not slip through.

The benefit is that the negative binomial distribution has closed-form formulas for every question a course or analyst is likely to ask, so a single deterministic calculator covers the whole workflow.

For continuous waiting-time problems with a constant rate λ, the Exponential Distribution Calculator is the continuous analogue of the geometric and negative binomial waiting-time families.

The shape of every output changes with r and p, and a few practical caveats apply when the underlying assumptions do not hold.

• • The calculator does not fit r and p from a sample, run a goodness-of-fit test, or simulate datasets.
• • The formulas assume independent Bernoulli trials with constant success probability p, so a drifting per-trial probability may not match the closed-form negative binomial.
• • For very small p the variance and skewness become very large, so floating-point precision can degrade for k larger than a few hundred failures.

Read the variance and the mean together whenever the result is sensitive to overdispersion. A variance much larger than the mean is the visual fingerprint of negative binomial data.

According to the Wikipedia article on the negative binomial distribution, the distribution is a gamma-Poisson mixture with variance μ/p and the standard overdispersed alternative to the Poisson distribution in regression and epidemiology.

According to the NIST Engineering Statistics Handbook entry on the Binomial Distribution, both the binomial and the negative binomial distribution rely on independent Bernoulli trials with constant success probability p, so the negative binomial inherits the same independence and constant-probability assumptions from the Bernoulli trial model.