The negative binomial distribution

### 8.4.4 The negative binomial distribution

#### The probability density function for the negative binomial distribution: negbinomial

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

| ⎛ ⎝ |
n+k−1
k
| ⎞ ⎠ | pn(1−p)k. (2)

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.

Input:

negbinomial(4,2,0.5)

Output:

0.15625

Note that

⎛ ⎝ |
n
k
| ⎞ ⎠ | = |
n!
k! (n−k)!
| = |
n (n−1 ) … ( n−k+1)
k!

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

negbinomial(n,k,p) = | ⎛ ⎝ |
−n
k
| ⎞ ⎠ | pn (p−1)k,

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: negbinomial_cdf

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,floor(x),p).

- negbinomial_cdf(n,p,x,y) returns

  Prob(x ≤ X ≤ y) = negbinomial(n,ceil(x),p) + ⋯ + negbinomial(n,floor(y),p)

#### Examples.

- Input:
  negbinomial_cdf(4,0.5,2)
  Output:

  0.34375

- Input:
  negbinomial_cdf(4,0.5,2,5)
  Output:

  0.40234375

#### The inverse distribution function for the negative binomial distribution: negbinomial_icdf

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.

Input:

negbinomial_icdf(4,0.5,0.9)

Output:

8