20.4.4  Negative binomial distribution

The probability density function for the negative binomial distribution.

If you repeatedly perform an experiment with probability of success p, then, given an integer n, the probability of k failures that occur before you have n successes is given by the negative binomial distribution, which can be computed by

The negbinomial command finds the density function for the negative binomial distribution.

• negbinomial takes three arguments:
  • n and k, integers.
  • p, a probability (a real number between 0 and 1).
• negbinomial(n,k,p) returns the value of the negative binomial distribution, given in (2).

Example

Note that

The second formula makes sense even if n is negative, and you can write

from which the name negative binomial distribution comes from. This also makes it simple to determine the mean (n(1−p)/p) and variance (n(1−p)/p²). The negative binomial is also called the Pascal distribution (after Blaise Pascal) or the Pólya distribution (after George Pólya).

The cumulative distribution function for the negative binomial distribution.

The negbinomial_cdf command finds the cumulative distribution function for the negative binomial distribution.

• negbinomial_cdf takes three mandatory arguments and two optional arguments:
  • n, an integer.
  • p, a probability (between 0 and 1).
  • x, a number.
  • Optionally, y, a number.
• negbinomial_cdf(n,p,x) returns

  Prob(X ≤ x)=negbinomial(n,0,p)+⋯+negbinomial(n,⌊ x⌋,p).

• negbinomial_cdf(n,p,x,y) returns

  Prob(x ≤ X ≤ y)=negbinomial(n,⌈ x⌉,p)+⋯ + negbinomial(n,⌊ y⌋,p)

Examples

The inverse distribution function for the negative binomial distribution.

The negbinomial_icdf command gives the inverse distribution function for the negative binomial distribution.

• negbinomial_icdf takes three arguments:
  • n, a positive integer.
  • p, a probability (a real number between 0 and 1).
  • h, a real number between 0 and 1.
• negbinomial_icdf(n,p,h) returns the value of the inverse distribution for the negative binomial distribution with n and probability p; namely, the smallest value of x for which Prob(X ≤ x) ≥ h.

Example