The negative binomial distribution
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The negative binomial distribution describes the probability
of
successes in a
sequence of independent experiments each with likelihood of success of
that
arise before there are
failures. In this
interpretation
is a positive
integer, but the distributional definition can also be extended to real values
of
. Note: different
texts adopt slightly different definitions, e.g. with support starting at
not
and/or with
denoting
probability of failure rather than probability of success.
Distribution name
|
Negative
binomial distribution
|
Common notation
|

|
Parameters
|
=
number of failures ( )
=
probability of success in each experiment ( )
|
Support
|

|
Probability mass
function
|

If is non-integral
then is:

|
Cumulative distribution
function
|

|
Mean
|

|
Variance
|

|
Skewness
|

|
(Excess) kurtosis
|

|
Characteristic function
|

|
Other comments
|
The geometric distribution is the same as the
negative binomial distribution with parameter .
Its pdf and cdf are therefore:


For the special case where is
an integer the negative binomial distribution is also called the Pascal
distribution. The Poisson
distribution is also a limiting case of the negative binomial:

|
Nematrian web functions
Functions relating to the above distribution may be accessed
via the Nematrian
web function library by using a DistributionName of “negative
binomial”. For details of other supported probability distributions see here.
NAVIGATION LINKS
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