NB

Negative Binomial Distribution Calculator

PMF · CDF · Mean · Variance · Bar Chart · Two Parameterizations

Compute exact probabilities, CDF and statistics for the negative binomial (Pascal) distribution. Supports both the failures-before-r-th-success and trial-of-r-th-success parameterizations.

Quick Examples

Parameterization

Mode A — Failures X = failures before r-th success Mode B — Trial number Y = trial of r-th success (Pascal)

r — Successes needed

Integer 1 – 50

p — Prob. of success

0 < p < 1

k — Failures (Mode A)

Integer ≥ 0

k_max — Display limit

Auto or 1 – 100

Full probability table from k=0 to k_max. Use the Calculator tab first, or enter parameters below.

Run the calculator first, then switch to this tab.

k	P(X=k)	P(X≤k)	P(X>k)

What Is the Negative Binomial Distribution?

The negative binomial distribution is a fundamental discrete probability distribution that models count data arising from a sequence of independent Bernoulli trials. Unlike the binomial distribution — which counts successes in a fixed number of trials — the negative binomial distribution has a fixed target number of successes r and counts how many failures (or total trials) occur before that target is reached.

Formally, suppose each trial independently results in success with probability p and failure with probability 1 − p. The process continues until exactly r successes have been observed. The negative binomial distribution describes the randomness in how many failures accumulate along the way. Its probability mass function is:

P(X = k) = C(k+r−1, k) × p^r × (1−p)^k k = 0, 1, 2, ...

where X is the number of failures before the r-th success. This form is particularly natural for quality control, where r = "number of good items required" and k = "number of defective items that appear before meeting the quota."

Two Parameterizations Explained

Statisticians use two equivalent ways to parameterize the negative binomial distribution, which can cause confusion when comparing sources.

Mode A — Number of Failures (X)

X counts the number of failures before the r-th success. The support is k = 0, 1, 2, .... The PMF is P(X=k) = C(k+r−1, k) × p^r × (1−p)^k. The mean is r(1−p)/p and variance is r(1−p)/p². This parameterization is common in probability textbooks.

Mode B — Trial Number (Y, Pascal Distribution)

Y counts the trial on which the r-th success occurs. The support is n = r, r+1, r+2, .... The PMF is P(Y=n) = C(n−1, r−1) × p^r × (1−p)^(n−r). The mean is r/p and variance is r(1−p)/p². This form is known as the Pascal distribution and is natural when the question is "on which attempt does the r-th success occur?"

Relationship to the Geometric Distribution

When r = 1, the negative binomial reduces exactly to the geometric distribution. In Mode A with r = 1, P(X=k) = p × (1−p)^k for k = 0, 1, 2, ..., which is the geometric distribution measuring the number of failures before the first success. In Mode B with r = 1, P(Y=n) = p × (1−p)^(n−1) for n = 1, 2, 3, ..., the standard geometric distribution. You can verify this on the calculator by setting r = 1.

Mean, Variance, and Overdispersion

Parameterization	Mean	Variance	Std Dev
Mode A (failures X)	r(1−p)/p	r(1−p)/p²	√(r(1−p)/p²)
Mode B (trials Y)	r/p	r(1−p)/p²	√(r(1−p)/p²)

A key property is that Variance = Mean + Mean²/r. Since Mean²/r > 0, the variance always exceeds the mean. This is called overdispersion — in contrast to the Poisson distribution, where variance equals the mean. As r → ∞ (with mean held fixed), the negative binomial converges to a Poisson distribution.

Relationship to the Binomial Distribution

While both distributions involve Bernoulli trials, their roles are reversed. In the binomial distribution B(n, p), n is fixed and successes X are counted. In the negative binomial, r successes are required and failures (or trials) are counted. The same binomial coefficient C(k+r−1, k) appears in the PMF as C(k+r−1, r−1), reflecting the ways r−1 successes can be distributed among the first k+r−1 trials before the final r-th success on trial k+r.

Applications

Insurance and actuarial science: Modelling claim counts when policyholders are heterogeneous. If each policyholder's claim rate follows a Gamma distribution, the marginal count is negative binomial.
Ecology: Species abundance and insect population counts are typically aggregated (overdispersed), making the negative binomial a much better fit than Poisson.
Genomics / RNA-seq: DESeq2, edgeR and other differential expression tools use the negative binomial distribution to model read counts per gene, explicitly modelling overdispersion.
Epidemiology: COVID-19 spreading data showed strong overdispersion (k parameter in NB notation), indicating superspreader dynamics. The negative binomial fits such data far better than Poisson.
Reliability engineering: Time to the r-th failure of a repairable system follows a negative binomial when individual failure probability is constant.
Sports analytics: Goals scored per game in soccer, runs in cricket innings, and wickets in bowling spells all show overdispersed patterns well captured by the negative binomial.
Marketing and sales: Number of sales calls before r confirmed orders, number of ad impressions before r clicks.

Frequently Asked Questions

What is the negative binomial distribution?

The negative binomial distribution is a discrete probability distribution modelling the number of failures (Mode A) or the trial number (Mode B) until a fixed number r of successes occurs, in a series of independent Bernoulli trials each with success probability p. Its PMF is P(X=k) = C(k+r−1, k) × p^r × (1−p)^k. It generalises the geometric distribution (r=1) and approaches the Poisson distribution as r → ∞.

What is the difference between binomial and negative binomial?

Binomial: fixed number of trials n, random number of successes. Negative binomial: fixed number of required successes r, random number of failures or trials. Binomial asks "how many heads in 10 flips?" Negative binomial asks "how many flips until 3 heads occur?" Both use the same binomial coefficient, but the support and interpretation differ fundamentally.

What do r and p represent in the negative binomial distribution?

r is the target number of successes — a fixed positive integer (or in the continuous extension, any positive real number). p is the probability of success on a single trial, 0 < p < 1. The trials are independent and identically distributed. r controls the "shape" of the distribution: larger r makes it more symmetric and bell-shaped, smaller r makes it more right-skewed.

What is the Pascal distribution?

The Pascal distribution is the negative binomial distribution in Mode B parameterization, where Y = trial number of the r-th success. It is named after Blaise Pascal. P(Y=n) = C(n−1, r−1) × p^r × (1−p)^(n−r) for n = r, r+1, .... Its mean is r/p (the expected trial on which the r-th success occurs) and variance is r(1−p)/p². Mode B on this calculator implements the Pascal distribution directly.

When does the negative binomial reduce to the geometric distribution?

When r = 1, the negative binomial becomes the geometric distribution. In Mode A: P(X=k) = p × (1−p)^k, the geometric distribution on {0, 1, 2, ...} with mean (1−p)/p. In Mode B: P(Y=n) = p × (1−p)^(n−1), the geometric distribution on {1, 2, 3, ...} with mean 1/p. This calculator shows a notice when r=1 is entered.

What is overdispersion and why is the negative binomial used for it?

Overdispersion means the sample variance exceeds the sample mean — more variability than a Poisson model (variance = mean) allows. In ecology, insurance, and genomics, count data routinely shows this property. The negative binomial has Var = Mean + Mean²/r, always exceeding the mean. The parameter r controls the degree of overdispersion: small r = strong overdispersion, large r → Poisson. This flexibility makes the negative binomial the default choice for overdispersed count regression.

What are real-world applications of the negative binomial distribution?

Key applications include: (1) RNA-seq differential expression analysis — DESeq2 and edgeR use NB to model read count overdispersion; (2) Epidemiology — COVID-19 secondary infections showed superspreading consistent with NB(r≈0.1); (3) Insurance — aggregate claims from heterogeneous portfolios; (4) Ecology — species abundance and aggregated insect distributions; (5) Reliability engineering — failures before r successful operations; (6) Sports — goals per match, wickets per spell; (7) Marketing — ad impressions before r conversions.