ABSTRACT
Consider a sequence of independent Bernoulli trials with success probability p. The distribution of the random variable that represents the number of failures until the first success is called geometric distribution. Now, let X denote the number of failures until the rth success. The random variable X is called the negative binomial random variable with parameters p and r, and its pmf is given by
P (X = k|r, p) = P (observing k failures in the first k + r − 1 trials) × P (observing a success at the (k + r)th trial)
=
( r + k − 1
k
) pr−1(1− p)k × p.
