Dr. Hackney STA Solutions pg 52

Dr. Hackney STA Solutions pg 52 - 4-8Solutions Manual for...

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Unformatted text preview: 4-8Solutions Manual for Statistical Inference4.29 a.XY=RcosθRsinθ= cotθ. LetZ= cotθ. LetA1= (0,π),g1(θ) = cotθ,g-11(z) = cot-1z,A2= (π,2π),g2(θ) = cotθ,g-12(z) =π+ cot-1z. By Theorem 2.1.8fZ(z) =12π|-11 +z2|+12π|-11 +z2|=1π11 +z2,-∞< z <∞.b.XY=R2cosθsinθthen 2XY=R22 cosθsinθ=R2sin 2θ. Therefore2XYR=Rsin 2θ.SinceR=√X2+Y2then2XY√X2+Y2=Rsin 2θ. Thus2XY√X2+Y2is distributed as sin 2θwhichis distributed as sinθ. To see this let sinθ∼fsinθ. For the function sin 2θthe values ofthe function sinθare repeated over each of the 2 intervals (0,π) and (π,2π) . Thereforethe distribution in each of these intervals is the distribution of sinθ. The probability ofchoosing between each one of these intervals is12. Thusf2 sinθ=12fsinθ+12fsinθ=fsinθ.Therefore2XY√X2+Y2has the same distribution asY= sinθ. In addition,2XY√X2+Y2has thesame distribution asX= cosθsince sinθhas the same distribution as cosθ. To see this letconsider the distribution ofW= cosθand...
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