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Справка Scilab >> Statistics > Cumulated Distribution Functions > cdfgam


cumulative distribution function gamma distribution





five real vectors of the same size.

P,Q (Q=1-P)

The integral from 0 to X of the gamma density. Input range: [0,1].


The upper limit of integration of the gamma density. Input range: [0, +infinity). Search range: [0,1E300]


The shape parameter of the gamma density. Input range: (0, +infinity). Search range: [1E-300,1E300]


The rate parameter of the gamma density. Input range: (0, +infinity). Search range: (1E-300,1E300]


Calculates any one parameter of the gamma distribution given values for the others.

The gamma density is

                f(x,a,r) = \frac{1}{r^{-a}\Gamma(a)} \int_0^x t^{a-1} \exp\left(-rt\right) dt

where a is the shape and r is the rate.

Caution. As opposed to other technical computing languages, this function makes use of the rate parameter, and not the scale parameter. The rate parameter is linked to the scale parameter with the equation rate=1/scale.

Computation of parameters such as X, Shape or Rate involve a search for a value that produces the desired value of P. The search relies on the monotonicity of P with the other parameter.


In the following example, we compute the probability of the event x=0.1 for the Gamma distribution function with Shape=1.0 and Rate=1.0.

Shape = 0.1
Rate = 1.0
x = 0.1
// Expected : P = 0.8275518

In the following example, we compute the probability of the event x=0.1 and check that the search algorithms allows to consistently invert the function.

Shape = 0.1
Rate = 2.0
x = 0.3
[P,Q]    = cdfgam("PQ",x,Shape,Rate)
[X1]     = cdfgam("X",Shape,Rate,P,Q)
[Shape1] = cdfgam("Shape",Rate,P,Q,x)
[Rate1]  = cdfgam("Rate",P,Q,x,Shape)

In the following example, we draw the Gamma distribution function for various values of the Shape and Rate.

N = 1000;
x = linspace(0,20,N)';
Shape = [1 2 3 5 9];
Rate = 1 ./ [2 2 2 1 0.5];
C = ["red" "green" "blue" "cyan" "orange"];
lstr = [];
h = gcf();
for i = 1 : 5
  P = cdfgam("PQ",x,Shape(i)*ones(N,1),Rate(i)*ones(N,1));
  lstr(i) = msprintf("Shape=%s, Rate=%s",string(Shape(i)),string(Rate(i)));
  h.children.children(1).children.foreground = color(C(i));
xtitle("CDF of the Gamma distribution function","X","P");


Cumulative distribution function (P) is calculated directly by the code associated with:

DiDinato, A. R. and Morris, A. H. Computation of the incomplete gamma function ratios and their inverse. ACM Trans. Math. Softw. 12 (1986), 377-393.

From DCDFLIB: Library of Fortran Routines for Cumulative Distribution Functions, Inverses, and Other Parameters (February, 1994) Barry W. Brown, James Lovato and Kathy Russell. The University of Texas.

See also

  • cdfbet — cumulative distribution function Beta distribution
  • cdfbin — cumulative distribution function Binomial distribution
  • cdfchi — cumulative distribution function chi-square distribution
  • cdfchn — cumulative distribution function non-central chi-square distribution
  • cdff — cumulative distribution function Fisher distribution
  • cdffnc — cumulative distribution function non-central f-distribution
  • cdfnbn — cumulative distribution function negative binomial distribution
  • cdfnor — cumulative distribution function normal distribution
  • cdfpoi — cumulative distribution function poisson distribution
  • cdft — cumulative distribution function Student's T distribution
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