1STAT 702/J702 B.Habing Univ. of S.C. 1STAT 702/J702 October 24th, 2006-Lecture 17-Instructor: Brian HabingDepartment of StatisticsTelephone: 803-777-3578E-mail: [email protected] 702/J702 B.Habing Univ. of S.C. 2Today• Functions of Jointly Distributed Random Variables• Order StatisticsSTAT 702/J702 B.Habing Univ. of S.C. 3For the continuous case the joint pdfof U=g1(X, Y) and V=g2(X, Y) isf U,V(u, v ) = fX,Y(h1(u, v ), h2(u, v )) |J|where h1and h2are the inverse functions: x =h1(u, v ), y =h2(u, v )) And J is the JacobiandvdhdudhdvdhdudhJ2211=2STAT 702/J702 B.Habing Univ. of S.C. 4Example 2) X and Y are bivariatenormal with means 0, variances 1, and correlation = 0.Let Find the joint and marginal distributions.⎟⎠⎞⎜⎝⎛=+=xyyxr1-22tan and θSTAT 702/J702 B.Habing Univ. of S.C. 53.6.1 - Special Case 1 – ConvolutionIn general say Z=X+YWe can find a general formula for FZ(z)=P(Z<z) simply by finding the appropriate area under f (x,y). Taking the derivative then gives us the pdf.STAT 702/J702 B.Habing Univ. of S.C. 6Example) X and Y are exponential RVs with parameter λ.3STAT 702/J702 B.Habing Univ. of S.C. 73.6.1 - Special Case 2 – QuotientA general formula for the quotient Z=Y/X can also be derived by examining the CDF. To do this easily, note that if y/x≤z then if x>0 we have y ≤ xzand if x<0 then y ≥ xz.STAT 702/J702 B.Habing Univ. of S.C. 8Back to the earlier example)X and Y have joint p.d.f. f XY(x,y ) = 2 0 ≤x < y≤1U=X/YSTAT 702/J702 B.Habing Univ. of S.C. 93.7 – Order StatisticsLet X1, X2, …. Xnbe independent random variables with the same CDF FX(x).The values in order from lowest to smallest are the order statistics X(1), X(2), …. X(n).4STAT 702/J702 B.Habing Univ. of S.C. 10First consider the maximum U=X(n).Note that U≤uif and only if all of the Xi≤u.)()( uUPuFU≤=))()((1uXuXPn≤∩∩≤= L)()(1uXPuXPn≤≤= LSTAT 702/J702 B.Habing Univ. of S.C. 11Taking the derivative we get:)()(1uXPuXPn≤≤= LnXXXuFuFuF )]([)()( == L1)]()[()(−=nXXUuFunfufSTAT 702/J702 B.Habing Univ. of S.C. 12The minimum V=X(1)works similarly:nXVvFvF )](1[1)( −−=1)](1)[()(−−=nXXVvFvnfvf5STAT 702/J702 B.Habing Univ. of S.C. 13Similar logic helps to get the marginal p.d.f. for any of the order statistics (as you will show in the homework).knkXkkkXkXxFxFxfknknxfk−−−⋅−−=)](1)[()()!()!1(!)()()(1)()()(STAT 702/J702 B.Habing Univ. of S.C. 14Example 1) Say you conduct 10 independent tests of hypotheses. How small should the smallest p-value be for you to reject it at a 0.05 level? That is, what is the 5th-%ile for the 1storder
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