gamcdf

Gamma cumulative distribution function

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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.

example

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.

example

[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)%.

example

___ = 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

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Compute Gamma Distribution cdf

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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

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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)

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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

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`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]