Beta distribution

The beta distribution depends on two parameters, a>0 and b>0; the value of the density function at x in [0,1] is (see Section 7.3.13):

betad(a,b,x)=

Γ(a+b)x^a−1(1−x)^b−1

Γ(a)Γ(b)

. (9)

The betad command computes the density function for the beta distribution.

betad takes three arguments:
- a and b, positive numbers, the parameters.
- x, a real number.
betad(a,b,x) returns the value of the density function for the beta distribution with parameters a and b, given in (9).

betad(2,1,0.3)

0.6

The betad_cdf command computes the cumulative distribution function for the beta distribution.

beta_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.
betad_cdf(a,b,x) returns Prob(X ≤ x) for the beta distribution with parameters a and b.
beta_cdf(n,x,y) returns Prob(x ≤ X ≤ y).

It turns out that betad_cdf(a,b,x)=β(a,b,x)Γ(a+b)/Γ(a)Γ(b), where β(a,b,x)=∫₀^x t^a−1(1−t)^b−1 dt (see Section 7.3.16).

betad_cdf(2,3,0.2)

0.1808

betad_cdf(2,3,0.25,0.5)

0.42578125

The betad_icdf command computes the inverse distribution for the beta distribution.

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

betad_icdf(2,3,0.2)

0.212317128278