9.4.12 The beta distribution
The probability density function for the beta distribution: betad
The beta distribution depends on two parameters, a>0 and b>0; the
value of the density function at x in [0,1] is
betad(a,b,x) = Γ(a+b)xa−1(1−x)b−1/(Γ(a)Γ(b))
(9) |
(see Section 6.8.13).
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).
Example.
Input:
betad(2,1,0.3)
Output:
The cumulative distribution function for the beta distribution: betad_cdf
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) = ∫0x ta−1(1−t)b−1 dt (see
Section 6.8.16).
Examples.
-
Input:
betad_cdf(2,3,0.2)
Output:
- Input:
betad_cdf(2,3,0.25,0.5)
Output:
The inverse distribution function for the beta distribution: betad_icdf
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.
Example.
Input:
betad_icdf(2,3,0.2)
Output: