# The gamma distribution

### 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−1e−bxba/Γ(a) (8)

The gammad command computes this density function.

- gammad takes three arguments:
  - a and b, positive real numbers (the parameters).
  - x, a real number.
- gammad(a,b,x) returns the value of the gamma density function with parameters a and b, given in (8).

#### 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.

- gamma_cdf takes three mandatory arguments and one optional argument:
  - a and b, real numbers (the parameters).
  - x, a real number.
  - Optionally, y, a real number.
- gamma_cdf(a,b,x) returns Prob(X ≤ x) for the gamma distribution with parameters a and b.
- gamma_cdf(n,x,y) returns Prob(x ≤ X ≤ y).

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

e−tta−1dt/Γ(a).

#### Examples.

- Input:
  gammad_cdf(2,1,0.5)
  Output:

  0.090204010431

- Input:
  gammad_cdf(2,1,0.5,1.5)
  Output:

  0.351970589198

#### The inverse distribution function for the gamma distribution: gammad_icdf

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

- gamma_icdf takes three arguments:
  - a and b, numbers (the parameters).
  - h, a real number between 0 and 1.
- gamma_icdf(a,b,h) returns the inverse distribution for the gamma distribution with parameters a and b; namely, the value of x for which Prob(X ≤ x) = h.

#### Example.

Input:

gammad_icdf(2,1,0.5)

Output:

1.67834699002