Answer to Question #242554 in Statistics and Probability for PRS

Question #242554

Given that u=χ\chi2m and v=χ\chi n2 further let u and v be independent variables obtain the probability density of F=(U/M)/(V/N) where m and n are the df for U and V


1
Expert's answer
2021-10-14T11:56:23-0400

Let us first determine the marginal distribution of random variables U and V. For convenience, let c=m/2c=m/2 and d=n/2d=n/2. To compute the distribution of FF, we need to find the joint distribution of UU and VV.

Note that,

fU(u)=(1/(Γ(c)2c))uc1eu/2, u>0f_U(u)=(1/(\varGamma(c)*2^c))*u^{c-1}e^{-u/2},\space u\gt0 and

fV(v)=(1/(Γ(d)2d))vd1ev/2, v>0f_V(v)=(1/(\varGamma(d)*2^d))*v^{d-1}e^{-v/2},\space v\gt0

Since UU and VV are independent, their joint distribution is given as,

fU,V(u,v)=(1/Γ(c)Γ(d)2c+d)uc1vd1eu/2ev/2, u,v>0f_{U,V}(u,v)=(1/\varGamma(c)\varGamma(d)2^{c+d})*u^{c-1}v^{d-1}e^{-u/2}e^{-v/2},\space u,v\gt0

We will first find the distribution of the random variable U/VU/V by using the cdfcdf method given as,

FU/V(f)=P(U/Vf)=P(UfV).F_{U/V}(f)=P(U/V\leqslant f)=P(U\leqslant fV).

Now,

P(UfV)=(1/(Γ(c)Γ(d)2c+d))0(0fvuc1eu/2du)vd1ev/2dvP(U\leqslant fV)=(1/(\varGamma(c)\varGamma(d)2^{c+d}))*\int^\infin_0(\int^{fv}_0u^{c-1}e^{-u/2}du)v^{d-1}e^{-v/2}dv

fU/V(f)=FU/V(f)=(fc1/Γ(c)Γ(d))0uc+d1e((f+1)/2)vdvf_{U/V}(f)= F'_{U/V}(f)=(f^{c-1}/\varGamma(c)\varGamma(d))\int^\infin_0u^{c+d-1}e^{-((f+1)/2)v}dv

Observe that,

f(v)=(f+1)c+duc+d1e((f+1)/2)vf(v)=(f+1)^{c+d}u^{c+d-1}e^{-((f+1)/2)v} is the pdfpdfof a Gamma random variable with parameters, α=(c+d)\alpha=(c+d) and β=(f+1)/2\beta=(f+1)/2.

Since 0f(v)=1,\int^\infin_0f(v)=1,

fU/V(f)=(Γ(c+d)/Γ(c)Γ(d))(fc1/(f+1)c+d)f_{U/V}(f)=(\varGamma(c+d)/\varGamma(c)\varGamma(d))*(f^{c-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/nf)f_F(f)=f_{(U/m,V/n)}(f)=f_{(n/m)(U/V)}(f)=(m/n)f_{U/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=\varGamma((m+n)/2)m^{m/2}n^{n/2}f^{(m/2)-1}/\varGamma(m/2)\varGamma(n/2)(n+mf)^{(m+n)/2},\space f\gt0

Which is an FF distribution with mm numerator degrees of freedom and nn denominator degrees of freedom.


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