nbinpdf

Negative binomial probability density function

Syntax

y = nbinpdf(x,r,p)

Description

y = nbinpdf(x,r,p) returns the negative binomial probability density function (pdf), evaluated at the values in x, using the corresponding number of successes r and the probability of success in a single trial p.

Examples

Compute and Plot Negative Binomial Probability Density Function

Compute the probability density function (pdf) of the negative binomial distribution with the parameters r=3 and p=0.5 at integer values between –1 and 10.

x = -1:1:10;
r = 3;
p = 0.5;
y = nbinpdf(x,3,0.5);

Plot the pdf.

plot(x,y,"o")
grid on
ylabel("y")
xlabel("x")

The negative binomial pdf values are nonzero only for nonnegative integer values of x.

Input Arguments

x — Values at which to evaluate negative binomial pdf

nonnegative scalar | array of nonnegative scalars

Values at which to evaluate the negative binomial pdf, specified as a nonnegative scalar or an array of nonnegative scalars. If x < 0 or is noninteger, the function returns a zero value for p.

To evaluate the pdf at multiple values, specify x as an array. To evaluate the pdfs of multiple distributions, specify either r or p (or both) using arrays. If one or more of the input arguments x, r, and p are arrays, then the array sizes must be the same. In this case, nbinpdf expands each scalar input into a constant array of the same size as the array inputs. Each element in y is the pdf value of the distribution specified by the corresponding elements in r and p, evaluated at the corresponding element in x.

Data Types: single | double

r — Number of successes

positive scalar | array of positive scalars

Number of successes, specified as a positive scalar or an array of positive scalars.

To evaluate the pdf at multiple values, specify x as an array. To evaluate the pdfs of multiple distributions, specify either r or p (or both) using arrays. If one or more of the input arguments x, r, and p are arrays, then the array sizes must be the same. In this case, nbinpdf expands each scalar input into a constant array of the same size as the array inputs. Each element in y is the pdf value of the distribution specified by the corresponding elements in r and p, evaluated at the corresponding element in x.

Data Types: single | double

p — Probability of success

positive scalar in the range (0,1] | array of positive scalars in the range (0,1]

Probability of success, specified as a positive scalar in the range (0,1] or an array of positive scalars in the range (0,1]. The function returns p=0 when x is less than 0 or is not an integer.

To evaluate the pdf at multiple values, specify x as an array. To evaluate the pdfs of multiple distributions, specify either r or p (or both) using arrays. If one or more of the input arguments x, r, and p are arrays, then the array sizes must be the same. In this case, nbinpdf expands each scalar input into a constant array of the same size as the array inputs. Each element in y is the pdf value of the distribution specified by the corresponding elements in r and p, evaluated at the corresponding element in x.

Data Types: single | double

Output Arguments

y — Negative binomial pdf values

numeric scalar | numeric array

Negative binomial pdf values, returned as a numeric scalar or array. y is the same size as x, r, and p after any necessary scalar expansion. Each element in y is the pdf value of the distribution specified by the corresponding elements in r and p, evaluated at the corresponding element in x.

More About

Negative Binomial pdf

The negative binomial pdf is

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

for x ≥ 0, where q=1 – p. The probability y is zero when x is not an integer.

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

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

The simplest application of the negative binomial distribution 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.

For more information, see Negative Binomial Distribution.

Alternative Functionality

• nbinpdf is a function specific to the negative binomial distribution. Statistics and Machine Learning Toolbox™ also offers the generic function pdf, which supports various probability distributions. To use pdf, create a NegativeBinomialDistribution probability distribution object and pass the object as an input argument or specify the probability distribution name and its parameters. Note that the distribution-specific function nbinpdf is faster than the generic function pdf.

• Use the Probability Distribution Function Tool to create an interactive plot of the cumulative distribution function (cdf) or probability density function (pdf) for a probability distribution.

Extended Capabilities

C/C++ Code Generation
Generate C and C++ code using MATLAB® Coder™.

GPU Arrays

Accelerate code by running on a graphics processing unit (GPU) using Parallel Computing Toolbox™.

This function fully supports GPU arrays. For more information, see Run MATLAB Functions on a GPU (Parallel Computing Toolbox).

Version History

Introduced before R2006a