Joint Distribution of Random Variables: Functions, Order Statistics, and Special Cases, Study notes of Statistics

Download Joint Distribution of Random Variables: Functions, Order Statistics, and Special Cases and more Study notes Statistics in PDF only on Docsity! 1 STAT 702/J702 B.Habing Univ. of S.C. 1 STAT 702/J702 October 24th, 2006 -Lecture 17- Instructor: Brian Habing Department of Statistics Telephone: 803-777-3578 E-mail: [email protected] STAT 702/J702 B.Habing Univ. of S.C. 2 Today • Functions of Jointly Distributed Random Variables • Order Statistics STAT 702/J702 B.Habing Univ. of S.C. 3 For the continuous case the joint pdf of U=g1(X, Y) and V=g2(X, Y) is f U,V(u, v ) = fX,Y(h1(u, v ), h2(u, v )) |J| where h1 and h2 are the inverse functions: x =h1(u, v ), y =h2(u, v )) And J is the Jacobian dv dh du dh dv dh du dh J 22 11 = 2 STAT 702/J702 B.Habing Univ. of S.C. 4 Example 2) X and Y are bivariate normal with means 0, variances 1, and correlation = 0. Let Find the joint and marginal distributions. ⎟ ⎠ ⎞ ⎜ ⎝ ⎛=+= x yyxr 1-22 tan and θ STAT 702/J702 B.Habing Univ. of S.C. 5 3.6.1 - Special Case 1 – Convolution In general say Z=X+Y We 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. 6 Example) X and Y are exponential RVs with parameter λ.