Updated Discrete Probability Tool

Negative Binomial Calculator

Work with both definitions of the negative binomial distribution: failures before the r-th success and the trial of the r-th success. Compute PMF, CDF, tail probabilities, probability tables, mean, variance, MGF and PGF.

Failures Before r-th Success Trial Count of r-th Success CDF and Tail Probabilities Distribution Formulas and Moments

Interactive Negative Binomial Distribution Calculator

Use the tabs to switch between the failures parameterization, the trial-count parameterization, probability tables and a summary of distribution properties. This helps you see how the negative binomial distribution behaves under both common definitions.

Here X counts failures before the r-th success. X can take values k = 0, 1, 2, … and is often written X ∼ NB(r, p) under this parameterization.

Here T counts the trial index of the r-th success. T can take values t = r, r + 1, r + 2, … and is related to X by T = X + r.

This table uses the failures parameterization X ∼ NB(r, p). It lists probabilities and cumulative probabilities for k from 0 up to kmax.

This tab summarizes the main formulas for the negative binomial distribution and evaluates them at your chosen r and p.

What Negative Binomial Calculator Calculates

Use Negative Binomial Calculator to work with both definitions of the negative binomial distribution: failures before the r-th success and the trial of the r-th success.

Related tools in this category are listed under Math Calculators.

Negative Binomial Calculator Inputs

Negative Binomial Calculator uses Number of required successes r, Success probability p, Failures k before r-th success, Decimal places, Trial t of r-th success.

Negative Binomial Calculator Formula or Method

  • P(X ≤ k) = Σ j = 0 k C(j + r − 1, j) p r q j
  • P(X = k) = C(k + r − 1, k) p r q k , for k = 0, 1, 2, …
  • P(X ≥ k) = Σ j = k ∞ C(j + r − 1, j) p r q j = 1 − P(X ≤ k − 1)

Reading Negative Binomial Calculator Results

The results include P(X = k), P(X ≤ k), P(X ≥ k), Mean of X, Variance of X, Support of X.

Alongside Negative Binomial Calculator, use Binomial Distribution Calculator to compute binomial probabilities, cumulative probabilities, tail probabilities and summary statistics.

For a different result from Negative Binomial Calculator, Binomial Probability Calculator can compute exact binomial probabilities for a given number of trials n, success probability p and number of successes k.

Calculators Related to Negative Binomial Calculator

After checking Negative Binomial Calculator, use Geometric Distribution Calculator to compute probabilities for the geometric distribution using both definitions: failures before the first success and trial number of the first success. When you need to compute PDF, CDF, Z-scores, tail probabilities and inverse normal values for any normal distribution, use Normal Distribution Calculator. Poisson Distribution Calculator covers a separate task: compute Poisson probabilities for event counts in a fixed interval.

After checking Negative Binomial Calculator, use Probability Calculator to work with single events, combined events and conditional probability. When you need to analyze datasets with arithmetic mean, weighted mean, geometric mean, harmonic mean, trimmed mean, frequency and grouped mean, use Mean Calculator. Mean, Variance & Weighted Statistics Calculator covers a separate task: compute mean and variance together, compare two datasets, generate full summary statistics, or calculate weighted mean and weighted variance.

After checking Negative Binomial Calculator, use Sampling Distribution Calculator to explore sampling distributions for sample means, proportions and differences between means.

Frequently Asked Questions About Negative Binomial Calculator

It is used to work with both definitions of the negative binomial distribution: failures before the r-th success and the trial of the r-th success.

It uses Number of required successes r, Success probability p, Failures k before r-th success, Decimal places, Trial t of r-th success.

Negative Binomial Calculator uses P(X ≤ k) = Σ j = 0 k C(j + r − 1, j) p r q j.

It returns P(X = k), P(X ≤ k), P(X ≥ k), Mean of X, Variance of X, Support of X.

Check Number of required successes r, Success probability p, Failures k before r-th success first. Differences can also come from rounding, unit choices, or a different formula or rule set.