# nbinpdf

Negative binomial probability density function

## Syntax

Y = nbinpdf(X,R,P)

## Description

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

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

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

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

## Examples

collapse all

Compute the pdf of a negative binomial distribution with parameters R = 3 and p = 0.5.

x = (0:10);
y = nbinpdf(x,3,0.5);

Plot the pdf.

figure;
plot(x,y,'+')
xlim([-0.5,10.5])