Statistics - Beta Distribution

Statistics – Beta Distribution

The beta distribution represents steady chance distribution parametrized by two constructive form parameters, αα and ββ, which seem as exponents of the random variable x and management the form of the distribution.

Beta Distribution

Chance density perform

Chance density perform of Beta distribution is given as:

Components

f(x)=(xa)α1(bx)β1B(α,β)(ba)α+β1axb;α,β>0where B(α,β)=10tα1(1t)β1dtf(x)=(x−a)α−1(b−x)β−1B(α,β)(b−a)α+β−1a≤x≤b;α,β>0where B(α,β)=∫01tα−1(1−t)β−1dt

The place −

  • α,βα,β = form parameters.
  • a,ba,b = higher and decrease bounds.
  • B(α,β)B(α,β) = Beta perform.

Customary Beta Distribution

In case of getting higher and decrease bounds as 1 and 0, beta distribution is named the usual beta distribution. It’s pushed by following formulation:

Components

f(x)=xα1(1x)β1B(α,β)x1;α,β>0f(x)=xα−1(1−x)β−1B(α,β)≤x≤1;α,β>0

Cumulative distribution perform

Cumulative distribution perform of Beta distribution is given as:

Components

F(x)=Ix(α,β)=x0tα1(1t)β1dtB(α,β)0x1;p,β>0F(x)=Ix(α,β)=∫0xtα−1(1−t)β−1dtB(α,β)0≤x≤1;p,β>0

The place −

  • α,βα,β = form parameters.
  • a,ba,b = higher and decrease bounds.
  • B(α,β)B(α,β) = Beta perform.

It is usually referred to as incomplete beta perform ratio.