Negative Binomial Distribution Percent Point

From probability q, number of successes r, and success probability p, find the smallest integer k where the negative binomial cumulative probability reaches q. Shows CDF, probability mass, upper tail, mean, variance and a PMF bar chart.

Input

Enter probability q, number of successes r, and success probability p to find the smallest non-negative integer k where the negative binomial cumulative probability P(X≤k) reaches q.

Probability q (greater than 0 and less than 1)

The target cumulative probability. The smallest k satisfying P(X≤k)≥q is returned.

Number of successes r (integer of 1 or more)

The number of successes to reach. X counts the failures observed before r successes.

Success probability p (greater than 0 and less than 1)

The probability of success on a single trial.

Result

Percent point k for q=0.9, r=5, p=0.5

Cumulative P(X≤k)

0.91021729

Mass P(X=k)

0.04364014

Upper tail P(X＞k)

0.08978271

Mean

Variance

Probability mass function (PMF)

How it works

The negative binomial distribution models the number of failures X observed before reaching r successes in trials with success probability p. Its probability mass is P(X=k)=C(k+r-1, k)·p^r·(1-p)^k.
The cumulative distribution is P(X≤k)=I_p(r, k+1) using the regularized incomplete beta function. The percent point is the smallest non-negative integer k where this cumulative probability reaches q.
The mean is r(1-p)/p and the variance is r(1-p)/p².
The binomial coefficient C(n, k) is evaluated with log factorials (the log gamma function) to avoid overflow.

Reviews

Tell us what you think of this calculator.