Note: This page has been translated by MathWorks. Please click here

To view all translated materals including this page, select Japan from the country navigator on the bottom of this page.

To view all translated materals including this page, select Japan from the country navigator on the bottom of this page.

Negative binomial cumulative distribution function

`y = nbincdf(x,R,p)`

y = nbincdf(x,R,p,'upper')

`y = nbincdf(x,R,p)`

computes
the negative binomial cdf 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.

`y = nbincdf(x,R,p,'upper')`

returns the
complement of the negative binomial cdf at each value in `x`

,
using an algorithm that more accurately computes the extreme upper
tail probabilities.

The negative binomial cdf is

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

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, `nbincdf`

allows `R`

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

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

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

Was this topic helpful?