For k-1Cr-1 × pr-1 × (1-p)k-r. we have a term and definition in Statistics.

K-1cr-1 × Pr-1 × (1-p)K-r. (Statistics)

for k = r, r+1, r+2, . . . , and zero for k < r, because there must be at least r trials to have r successes. The negative binomial distribution is derived as follows: for the rth success to occur on the kth trial, there must have been r-1 successes in the first k-1 trials, and the kth trial must result in success. The chance of the former is the chance of r-1 successes in k-1 independent trials with the same probability of success in each trial, which, according to the Binomial distribution with parameters n=k-1 and p, has probability

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