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9.4.4  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.


Example.
Input:

negbinomial(4,2,0.5)

Output:

0.15625


Note that


n
 
k

n!
k! (nk)!
 = 
n (n−1 ) … ( nk+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.


Examples.

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.


Example.
Input:

negbinomial_icdf(4,0.5,0.9)

Output:

8

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