11-8Solutions Manual for Statistical InferenceFactor out the terms withynand do some algebra on the middle term to getf(y1, y2, . . . , yn)=yΣiλi-1ne-yn1Γ(λ1)1n-1j=1(1 +yj)λ1-1×n-1i=21Γ(λi)yi-11 +yi-11n-1j=i(1 +yj)λi-1×n-1i=11(1 +yi)n-1j=i(1 +yj).We see thatYnis independent of the otherYi(and has a gamma distribution), but theredoes not seem to be any other obvious conclusion to draw from this density.b. TheYiare related to theFdistribution in the ANOVA. For example, as long as the sumof theλiare integers,Yi=Xi+1∑ij=1Xj=2Xi+12∑ij=1Xj=χ2λi+1χ2∑ij=1λj∼Fλi+1,∑ij=1λj.Note that theFdensity makes sense even if theλiare not integers.11.21 a.Grand mean ¯y··=188.5415=12.57Total sum of squares=3i=15j=1(yij-¯y··)2=1295.01.Within SS=3151(yij-¯yi·)2=51(y1j-3.508)
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