Let us first determine the marginal distribution of random variables U and V. For convenience, let c=m/2 and d=n/2. To compute the distribution of F, we need to find the joint distribution of U and V.
Note that,
fU(u)=(1/(Γ(c)∗2c))∗uc−1e−u/2, u>0 and
fV(v)=(1/(Γ(d)∗2d))∗vd−1e−v/2, v>0
Since U and V are independent, their joint distribution is given as,
fU,V(u,v)=(1/Γ(c)Γ(d)2c+d)∗uc−1vd−1e−u/2e−v/2, u,v>0
We will first find the distribution of the random variable U/V by using the cdf method given as,
FU/V(f)=P(U/V⩽f)=P(U⩽fV).
Now,
P(U⩽fV)=(1/(Γ(c)Γ(d)2c+d))∗∫0∞(∫0fvuc−1e−u/2du)vd−1e−v/2dv
fU/V(f)=FU/V′(f)=(fc−1/Γ(c)Γ(d))∫0∞uc+d−1e−((f+1)/2)vdv
Observe that,
f(v)=(f+1)c+duc+d−1e−((f+1)/2)v is the pdfof a Gamma random variable with parameters, α=(c+d) and β=(f+1)/2.
Since ∫0∞f(v)=1,
fU/V(f)=(Γ(c+d)/Γ(c)Γ(d))∗(fc−1/(f+1)c+d)
We now have that,
fF(f)=f(U/m,V/n)(f)=f(n/m)(U/V)(f)=(m/n)fU/V(m/n∗f)
=Γ((m+n)/2)mm/2nn/2f(m/2)−1/Γ(m/2)Γ(n/2)(n+mf)(m+n)/2, f>0
Which is an F distribution with m numerator degrees of freedom and n denominator degrees of freedom.
Comments
Leave a comment