Editing Negative binomial distribution (section)

===Waiting time in a Bernoulli process===

Let {{mvar|k}} and {{mvar|r}} be integers with {{mvar|k}} non-negative and {{mvar|r}} positive.  In a sequence of independent [[Bernoulli trial]]s with success probability {{mvar|p}}, the negative binomial gives the probability of {{mvar|k}} successes and {{mvar|r}} failures, with a failure on the last trial. Therefore, the negative binomial distribution represents the probability distribution of the number of successes before the {{mvar|r}}-th failure in a [[Bernoulli process]], with probability {{mvar|p}} of successes on each trial.

Consider the following example. Suppose we repeatedly throw a die, and consider a 1 to be a failure.  The probability of success on each trial is 5/6. The number of successes before the third failure belongs to the infinite set {{math|{{mset| 0, 1, 2, 3, ... }}}}. That number of successes is a negative-binomially distributed random variable.

When {{math|1=''r'' = 1}} we get the probability distribution of number of successes before the first failure (i.e. the probability of the first failure occurring on the {{math|(''k'' + 1)}}-st trial), which is a [[geometric distribution]]:
: <math>
    f(k; r, p) = (1-p) \cdot p^k \!
  </math>