Negative Binomial Distribution
Calculate negative binomial distribution (Pascal distribution) probabilities, mean, variance, and standard deviation. Free online calculator with PMF and CDF charts.
About This Calculator
The Negative Binomial Distribution calculator (also known as the Pascal distribution calculator) helps you compute the probability of requiring a specific number of trials to achieve a fixed number of successes. This distribution is essential in quality control, epidemiology, sports analytics, and risk assessment where you need to model the number of attempts until a predetermined number of events occur.
The probability mass function is P(Y = n) = C(n-1, r-1) × p^r × (1-p)^(n-r), where n is the total number of trials, r is the required number of successes, and p is the probability of success on each trial. The mean is μ = r/p and the variance is σ² = r(1-p)/p².
Regional Notes
India: Used in industrial quality control (ISI/ISO standards) for number of defects until a sample is rejected, and in epidemiological studies for disease cluster detection across Indian states.
United States: Widely used in FDA medical device trials to determine sample sizes for rare event testing, in baseball sabermetrics for streak analysis, and in Six Sigma DMAIC projects for defect counting until process improvement.
United Kingdom: Applied in actuarial science by UK insurance companies for modeling the number of claims until a threshold is reached, and in clinical trial design under MHRA guidelines for adaptive trial designs with rare endpoints.
Frequently Asked Questions
What is the negative binomial distribution?
The negative binomial distribution (also called the Pascal distribution) models the probability of needing n trials to achieve r successes, where each trial has a constant success probability p. The probability mass function is P(Y = n) = C(n-1, r-1) × p^r × (1-p)^(n-r) for n = r, r+1, r+2, ...
What is the difference between binomial and negative binomial distribution?
In a binomial distribution, the number of trials is fixed and the random variable is the number of successes. In a negative binomial distribution, the number of successes is fixed and the random variable is the number of trials needed to achieve those successes. The negative binomial has possible values from r to infinity, while the binomial ranges from 0 to n.
What are real-world examples of the negative binomial distribution?
Common examples include: the number of times you need to roll a die to get three 6s, the number of door knocks needed to sell 10 subscriptions, the number of attempts required to score 3 goals in a match, or the number of patients you need to examine to find 5 cases of a rare disease.
What is the mean of a negative binomial distribution?
The mean (expected value) of a negative binomial distribution is μ = r/p, where r is the number of successes and p is the probability of success per trial. For example, if you need 15 successes with a 40% success rate, the expected number of trials is 15/0.4 = 37.5.
What is the variance of a negative binomial distribution?
The variance of a negative binomial distribution is σ² = r(1-p)/p², where r is the number of successes and p is the probability of success. The standard deviation is the square root of this value. The variance increases as the success probability decreases, reflecting greater uncertainty with rarer events.
What is the difference between negative binomial and geometric distribution?
The geometric distribution is a special case of the negative binomial distribution where r = 1. The geometric distribution models the number of trials needed for the first success, while the negative binomial generalizes this to any fixed number of successes. Both share the same underlying process of Bernoulli trials with constant success probability.
Can the number of trials n be any positive integer?
Yes, n must be an integer greater than or equal to r (the required number of successes). Unlike the binomial distribution which has a fixed upper bound, the negative binomial distribution has no upper limit — n can theoretically go to infinity, though probabilities become very small for large n. Our calculator shows probabilities for n up to r + 50 or 5 standard deviations above the mean.