gamcdf

Gamma cumulative distribution function

Syntax

p = gamcdf(x,a)

p = gamcdf(x,a,b)

[p,pLo,pUp] = gamcdf(x,a,b,pCov)

[p,pLo,pUp] = gamcdf(x,a,b,pCov,alpha)

___ = gamcdf(___,'upper')

Description

p = gamcdf(x,a) returns the cumulative distribution function (cdf) of the standard gamma distribution with the shape parameters in a, evaluated at the values in x.

p = gamcdf(x,a,b) returns the cdf of the gamma distribution with the shape parameters in a and scale parameters in b, evaluated at the values in x.

[p,pLo,pUp] = gamcdf(x,a,b,pCov) also returns the 95% confidence interval [pLo,pUp] of p when a and b are estimates. pCov is the covariance matrix of the estimated parameters.

[p,pLo,pUp] = gamcdf(x,a,b,pCov,alpha) specifies the confidence level for the confidence interval [pLo pUp] to be 100(1–alpha)%.

___ = gamcdf(___,'upper') returns the complement of the cdf, evaluated at the values in x, using an algorithm that more accurately computes the extreme upper-tail probabilities than subtracting the lower tail value from 1. 'upper' can follow any of the input argument combinations in the previous syntaxes.

Examples

Compute Gamma Distribution cdf

Compute the cdf of the mean of the gamma distribution, which is equal to the product of the parameters ab.

a = 1:6;
b = 5:10;
prob = gamcdf(a.*b,a,b)

prob = 1×6

    0.6321    0.5940    0.5768    0.5665    0.5595    0.5543

As ab increases, the distribution becomes more symmetric, and the mean approaches the median.

Confidence Interval of Gamma cdf Value

Find a confidence interval estimating the probability that an observation is in the interval [0 10] using gamma distributed data.

Generate a sample of 1000 gamma distributed random numbers with shape 2 and scale 5.

x = gamrnd(2,5,1000,1);

Compute estimates for the parameters.

[params,~] = gamfit(x)

params = 1×2

    2.1089    4.8147

Store the parameters as ahat and bhat.

ahat = params(1);
bhat = params(2);

Find the covariance of the parameter estimates.

[~,nCov] = gamlike(params,x)

nCov = 2×2

    0.0077   -0.0176
   -0.0176    0.0512

Create a confidence interval estimating the probability that an observation is in the interval [0 10].

[prob,pLo,pUp] = gamcdf(10,ahat,bhat,nCov)

prob = 0.5830

pLo = 0.5587

pUp = 0.6069

Complementary cdf (Tail Distribution)

Determine the probability that an observation from the gamma distribution with shape parameter 2 and scale parameter 3 will is in the interval [150 Inf].

p1 = 1 - gamcdf(150,2,3)

p1 = 0

gamcdf(150, 2, 3) is nearly 1, so p1 becomes 0. Specify 'upper' so that gamcdf computes the extreme upper-tail probabilities more accurately.

p2 = gamcdf(150,2,3,'upper')

p2 = 9.8366e-21

Input Arguments

x — Values at which to evaluate cdf

nonnegative scalar value | array of nonnegative scalar values

Values at which to evaluate the cdf, specified as a nonnegative scalar value or an array of nonnegative scalar values.

If you specify pCov to compute the confidence interval [pLo,pUp], then x must be a scalar value.

• To evaluate the cdf at multiple values, specify x using an array.

• To evaluate the cdfs of multiple distributions, specify a and b using arrays.

If one or more of the input arguments x, a, and b are arrays, then the array sizes must be the same. In this case, gamcdf expands each scalar input into a constant array of the same size as the array inputs. Each element in p is the cdf value of the distribution specified by the corresponding elements in a and b, evaluated at the corresponding element in x.

Example: [3 4 7 9]

Data Types: single | double

a — Shape parameter

positive scalar value | array of positive scalar values

Shape of the gamma distribution, specified as a positive scalar value or an array of positive scalar values.

• To evaluate the cdf at multiple values, specify x using an array.

• To evaluate the cdfs of multiple distributions, specify a and b using arrays.

If one or more of the input arguments x, a, and b are arrays, then the array sizes must be the same. In this case, gamcdf expands each scalar input into a constant array of the same size as the array inputs. Each element in p is the cdf value of the distribution specified by the corresponding elements in a and b, evaluated at the corresponding element in x.

Example: [1 2 3 5]

Data Types: single | double

b — Scale parameter

1 (default) | positive scalar value | array of positive scalar values

Scale of the gamma distribution, specified as a positive scalar value or an array of positive scalar values.

• To evaluate the cdf at multiple values, specify x using an array.

• To evaluate the cdfs of multiple distributions, specify a and b using arrays.

If one or more of the input arguments x, a, and b are arrays, then the array sizes must be the same. In this case, gamcdf expands each scalar input into a constant array of the same size as the array inputs. Each element in p is the cdf value of the distribution specified by the corresponding elements in a and b, evaluated at the corresponding element in x.

Example: [1 1 2 2]

Data Types: single | double

pCov — Covariance of estimates

2-by-2 numeric matrix

Covariance of the estimates a and b, specified as a 2-by-2 matrix.

If you specify pCov to compute the confidence interval [pLo,pUp], then x, a, and b must be scalar values.

You can estimate a and b by using gamfit or mle, and estimate the covariance of a and b by using gamlike. For an example, see Confidence Interval of Gamma cdf Value.

Data Types: single | double

alpha — Significance level

0.05 (default) | scalar in the range (0,1)

Significance level for the confidence interval, specified as a scalar in the range (0,1). The confidence level is 100(1–alpha)%, where alpha is the probability that the confidence interval does not contain the true value.

Example: 0.01

Data Types: single | double

Output Arguments

p — cdf values

scalar value | array of scalar values

cdf values evaluated at the values in x, returned as a scalar value or an array of scalar values. p is the same size as x, a, and b after any necessary scalar expansion. Each element in p is the cdf value of the distribution specified by the corresponding elements in a and b, evaluated at the corresponding element in x.

pLo — Lower confidence bound for p

scalar value | array of scalar values

Lower confidence bound for p, returned as a scalar value or an array of scalar values. pLo has the same size as p.

pUp — Upper confidence bound for p

scalar value | array of scalar values

Upper confidence bound for p, returned as a scalar value or an array of scalar values. pUp has the same size as p.

More About

Gamma cdf

The gamma distribution is a two-parameter family of curves. The parameters a and b are shape and scale, respectively.

The gamma cdf is

$$p=F(x|a,b)=\frac{1}{b^a\Gamma (a)}{\displaystyle \underset{0}{\overset{x}{\int }}{t}^{a-1}{e}^{\frac{-t}{b}}dt}.$$

The result p is the probability that a single observation from a gamma distribution with parameters a and b falls in the interval [0,x].

The gamma cdf is related to the incomplete gamma function gammainc by

$$f\left(x|a,b\right)=\text{gammainc}\left(\frac{x}{b},a\right).$$

The standard gamma distribution occurs when b = 1, which coincides with the incomplete gamma function precisely.

For more information, see Gamma Distribution.

Alternative Functionality

• gamcdf is a function specific to the gamma distribution. Statistics and Machine Learning Toolbox™ also offers the generic function cdf, which supports various probability distributions. To use cdf, create a GammaDistribution 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 gamcdf is faster than the generic function cdf.

• Use the Probability Distribution Function app 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).