numpy.random.negative_binomial(n, p, size=None)

Draw samples from a negative binomial distribution.

Samples are drawn from a negative binomial distribution with specified parameters, n trials and p probability of success where n is an integer > 0 and p is in the interval [0, 1].


n : int

Parameter, > 0.

p : float

Parameter, >= 0 and <=1.

size : int or tuple of ints, optional

Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. Default is None, in which case a single value is returned.


samples : int or ndarray of ints

Drawn samples.


The probability density for the negative binomial distribution is

P(N;n,p) = \binom{N+n-1}{n-1}p^{n}(1-p)^{N},

where n-1 is the number of successes, p is the probability of success, and N+n-1 is the number of trials. The negative binomial distribution gives the probability of n-1 successes and N failures in N+n-1 trials, and success on the (N+n)th trial.

If one throws a die repeatedly until the third time a “1” appears, then the probability distribution of the number of non-“1”s that appear before the third “1” is a negative binomial distribution.


[R243]Weisstein, Eric W. “Negative Binomial Distribution.” From MathWorld–A Wolfram Web Resource.
[R244]Wikipedia, “Negative binomial distribution”,


Draw samples from the distribution:

A real world example. A company drills wild-cat oil exploration wells, each with an estimated probability of success of 0.1. What is the probability of having one success for each successive well, that is what is the probability of a single success after drilling 5 wells, after 6 wells, etc.?

>>> s = np.random.negative_binomial(1, 0.1, 100000)
>>> for i in range(1, 11):
...    probability = sum(s<i) / 100000.
...    print i, "wells drilled, probability of one success =", probability