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, and can be computed with negbinomial(n,k,p). It is given by the formula (kn+k−1)pn(1−p)k. If you enter
you will get
Note that
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The second formula makes sense even if n is negative, and you can write negbinomial(n,k,p) = (k−n)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 is given by the negbinomial_cdf command. Given parameters n and p, as above, then negbinomial_cdf(n,p,x) = Prob(X ≤ x) = negbinomial(n,0,p) + …+ negbinomial(n,floor(x),p), and negbinomial_cdf(n,p,x,y) = Prob(x ≤ X ≤ y) = negbinomial(n,ceil(x),p) + ⋯ + negbinomial(n,floor(y),p). If you enter
for example, you will get
Given a value h, the inverse distribution function gives the smallest value of x so that Prob(X ≤ x) ≥ h. The negbinomial_icdf gives the inverse distribution function for the negative binomial distribution. If you enter
for example, you will get