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Unformatted text preview: 4-8Solutions Manual for Statistical Inference4.29 a.XY=RcosRsin= cot. LetZ= cot. LetA1= (0,),g1() = cot,g-11(z) = cot-1z,A2= (,2),g2() = cot,g-12(z) =+ cot-1z. By Theorem 2.1.8fZ(z) =12|-11 +z2|+12|-11 +z2|=111 +z2,-< z <.b.XY=R2cossinthen 2XY=R22 cossin=R2sin 2. Therefore2XYR=Rsin 2.SinceR=X2+Y2then2XYX2+Y2=Rsin 2. Thus2XYX2+Y2is distributed as sin 2whichis distributed as sin. To see this let sinfsin. For the function sin 2the values ofthe function sinare repeated over each of the 2 intervals (0,) and (,2) . Thereforethe distribution in each of these intervals is the distribution of sin. The probability ofchoosing between each one of these intervals is12. Thusf2 sin=12fsin+12fsin=fsin.Therefore2XYX2+Y2has the same distribution asY= sin. In addition,2XYX2+Y2has thesame distribution asX= cossince sinhas the same distribution as cos. To see this letconsider the distribution ofW= cosand...
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