Derivation of Beta Distribution from F-Distribution
📂Probability DistributionDerivation of Beta Distribution from F-Distribution
Theorem
A random variable X∼F(r1,r2) following an F-distribution with degrees of freedom r1,r2 is defined as follows Y and follows a beta distribution Best(2r1,2r2).
Y:=1+(r1/r2)X(r1/r2)X∼Beta(2r1,2r2)
Proof
Strategy: Direct deduction using the probability density function.
Definition of F-distribution: A continuous probability distribution F(r1,r2) with the following probability density function for degrees of freedom r1,r2>0 is called an F-distribution.
f(x)=B(r1/2,r2/2)1(r2r1)r1/2xr1/2−1(1+r2r1x)−(r1+r2)/2,x∈(0,∞)
Definition of Beta Distribution: A continuous probability distribution Beta(α,β) with the following probability density function for α,β>0 is called a Beta distribution.
f(x)=B(α,β)1xα−1(1−x)β−1,x∈[0,1]
- B(r1/2,r2/2) signifies the Beta function.
⟹⟹⟹Y=1+(r1/r2)X(r1/r2)XY(1+(r1/r2)X)=(r1/r2)XY=(r1/r2)X(1−Y)(r1/r2)X=1−YY
and
dy===[1+(r1/r2)x(r1/r2)−(r1/r2)[1+(r1/r2)x]2(r1/r2)x]dx1+(r1/r2)x(r1/r2)[1+(r1/r2)x1+(r1/r2)x−1+(r1/r2)x(r1/r2)x]dx[1+(r1/r2)x]2(r1/r2)dx
therefore, the probability density function fY of Y is
========B(r1/2,r2/2)fY(y)(r2r1)r1/2xr1/2−1(1+r2r1x)−(r1+r2)/2⋅(r1/r2)[1+(r1/r2)x]2(r2r1)r1/2−1xr1/2−1(1+r2r1x)2−(r1+r2)/2(r2r1x)r1/2−1(1+r2r1x)2−(r1+r2)/2yr1/2−1(1+r2r1x)r1/2−1(1+r2r1x)2−(r1+r2)/2yr1/2−1(1+r2r1x)1−r2/2yr1/2−1(1+1−yy)1−r2/2yr1/2−1(1−y1)1−r2/2yr1/2−1(1−y)r2/2−1
Summarizing, Y has the following probability density function Beta(2r1,2r2).
fY(y)=B(r1/2,r2/2)1yr1/2−1(1−y)r2/2−1
■