The Negative Binomial Distribution
The negative binomial distribution has the following requirements:
A sequence of independent trials.
Each trial can result in success (S) or failure (F).
The probability of success,
is a constant.Trials continue until r successes have been obtained.
The random variable of interest,
is the number of failures that precede the
th success.
may take any value greater than or equal to 0. Let
denote the probability mass function of
The event
is equivalent to
S's in the first
trials and an S on the (
)th trial. If
and
then there must be four S's in the first 14 trials and the fifteenth trial must be an S. Since the trials are independent,
The first factor here is only the binomial probability of gettingS's intrials: 
Multiplying this by
gives
Like the normal binomial expansion, the negative binomial distribution is well defined even when r is not an integer. In this case we have the generalized negative binomial distribution.
The mean and variance are given by
and
Example: A paediatrician wishes to recruit five couples, each expecting their first child. Let
be the probability that the couple agree to be recruited. What is the probability that fifteen couples are asked before the required five agree?
We have
and
