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Negative Binomial Distribution
Consider a statistical experiment where a success occurs with
probability p and a failure occurs with probability q
= 1 - p. If the experiment is repeated indefinitely and
the trials are independent of each other, then the random
variable X, whose value is the number of the trial on which the rth
success occurs, has a negative binomial distribution with
parameters r and p. The probability mass
function of X is
For the rth success to occur on the xth trial, there
must have been (r-1) successes and (x-r)
failures among the first (x-1) trials. The number of ways of
distributing (r-1)
successes among (x - 1) trials is  The probability of
having ( -1)
successes and (x - r) failures is p r-1(1
- p)x-r
The probability of the rth success is p.
Thus, the product of these three terms is the probability that
there are r successes and x-r failures in the x
trials, with the rth success occurring on the xth trial.
A random variable X, having a negative binomial distribution
with parameters r and p, is the sum of r
independent random variables, each one geometrically distributed
with parameter p. Intuitively, X is the number of trials
needed for the first success, plus the number of trials needed
for the second success, ........ , plus the number of trials
needed for the rth success. Thus, the mean and variance
of a random variable X, with parameters r and p,
are derived as follows:
In fact, a geometric distribution with parameter p is
the same as a negative binomial distribution with parameters n
= 1 and p.
EX. A phenomenal major-league baseball player has a batting
average of 0.400. Beginning with his next at-bat, the random
variable X, whose value refers to the number of the at-bat
(walks, sacrifice flies and certain types of outs are not
considered at-bats) when his rth hit occurs, has a
negative binomial distribution with parameters r and p
= 0.400. It has the following probability mass function:
The probability that this hitter's second hit comes on the
fourth at-bat is
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