Negative binomial probability density function

`Y = nbinpdf(X,R,P)`

`Y = nbinpdf(X,R,P)`

returns
the negative binomial pdf at each of the values in `X`

using
the corresponding number of successes, `R`

and probability
of success in a single trial, `P`

. `X`

, `R`

,
and `P`

can be vectors, matrices, or multidimensional
arrays that all have the same size, which is also the size of `Y`

.
A scalar input for `X`

, `R`

, or `P`

is
expanded to a constant array with the same dimensions as the other
inputs. Note that the density function is zero unless the values in `X`

are
integers.

The negative binomial pdf is

$$y=f(x|r,p)=\left(\begin{array}{c}r+x-1\\ x\end{array}\right){p}^{r}{q}^{x}{I}_{(0,1,\mathrm{...})}(x)$$

The simplest motivation for the negative binomial is the case
of successive random trials, each having a constant probability `P`

of
success. The number of *extra* trials you must
perform in order to observe a given number `R`

of
successes has a negative binomial distribution. However, consistent
with a more general interpretation of the negative binomial, `nbinpdf`

allows `R`

to
be any positive value, including nonintegers. When `R`

is
noninteger, the binomial coefficient in the definition of the pdf
is replaced by the equivalent expression

$$\frac{\Gamma (r+x)}{\Gamma (r)\Gamma (x+1)}$$