5-8Solutions Manual for Statistical Inference=1√2πνν/2Γ(ν/2)2ν/2∞0x((ν+1)/2)-1e-(ν+t2)x/2dxintegrand is kernel ofgamma((ν+1)/2,2/(ν+t2)=1√2πνν/2Γ(ν/2)2ν/2Γ((ν+ 1)/2)2ν+t2(ν+1)/2=1√νπΓ((ν+1)/2)Γ(ν/2)1(1 +t2/ν)(ν+1)/2,the pdf of atνdistribution.b. Differentiate both sides with respect totto obtainνfF(νt) =∞0yf1(ty)fν(y)dy,wherefFis theFpdf. Now write out the two chi-squared pdfs and collect terms to getνfF(νt)=t-1/2Γ(1/2)Γ(ν/2)2(ν+1)/2∞0y(ν-1)/2e-(1+t)y/2dy=t-1/2Γ(1/2)Γ(ν/2)2(ν+1)/2Γ(ν+12)2(ν+1)/2(1 +t)(ν+1)/2.Now definey=νtto getfF(y) =Γ(ν+12)νΓ(1/2)Γ(ν/2)(y/ν)-1/2(1 +y/ν)(ν+1)/2,the pdf of anF1,ν.c. Again differentiate both sides with respect tot, write out the chi-squared pdfs, and collectterms to obtain(ν/m)fF((ν/m)t) =t-m/2Γ(m/2)Γ(ν/2)2(ν+m)/2∞0y(m+ν-2)/2e-(1+t)y/2dy.Now, as before, integrate the gamma kernel, collect terms, and definey= (ν/m)tto getfF(
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