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8.4.11 The gamma distribution

The probability density function for the gamma distribution: gammad

The gamma distribution depends on two parameters, a>0 and b>0; the value of the density function at x ≥ 0 is

gammad(a,b,x) = xa−1ebxba/Γ(a) (8)

The gammad command computes this density function.

Example

Input:

gammad(2,1,3)

Output:

3
e3

The cumulative distribution function for the gamma distribution: gammad_cdf

The gamma_cdf command computes the cumulative distribution function for the gamma distribution.

It turns out that

gammad_cdf(n,x) = igamma(a, bx, 1)

where igamma is the incomplete gamma function (see Section ‍5.8.15),

  igamma(a,x,1) = 
x


0
etta−1dt/Γ(a).

Examples

The inverse distribution function for the gamma distribution: gammad_icdf

The gammad_icdf command computes the inverse distribution for the gamma distribution.

Example

Input:

gammad_icdf(2,1,0.5)

Output:

1.67834699002

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