INTRO STAT THEORY I
INTRO STAT THEORY I STAT 702
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This 4 page Class Notes was uploaded by Shane Marks on Monday October 26, 2015. The Class Notes belongs to STAT 702 at University of South Carolina - Columbia taught by Staff in Fall. Since its upload, it has received 29 views. For similar materials see /class/229677/stat-702-university-of-south-carolina-columbia in Statistics at University of South Carolina - Columbia.
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Date Created: 10/26/15
STAT 702J702 October 19th 2004 Lecfure 77 lnstructor Brian Habing Department of Statistics Telephone 8037773578 Email habingstatscedu V4 STAT 702J702 BHabing Univ ofSC Today Homework Solution Functions of Joint Random Variables continued STAT 702J702 BHabing Univ ofSC w Homework 11 Consider the joint pdf fxy1a12x12y defined forO lt xy lt 1 1 lt 3 lt1 Find what conditions must be met for X and Y to be independent STAT 702J702 BHabing Univ ofSC w 362 Functions of Joint Random Variables X Y have joint pdf fXYx y We want the distribution of Ug1X Y V92X Y Eg UXY VY2 or UXY VX2Y etc STAT 702J702 BHabing Univ ofSC M For the continuous case the joint pdf of UN is fUvU1 V 7 ltY71U V h2U V IJI where I71 and I72 are the inverse functions xh1u V yh2u V And J is the Jacobian 1 dquot W du dv STAT 702J702 BHabing Univ ofSC w Example 1 X and Y havejoint pdf fXYXy 2 Osxltyg1 UXY and VY Find the joint and marginal pdf s of X and Y STAT 702J702 BHabing Univ ofSC w Example 2 X and Y are bivariate normal with means 0 variances 1 and correlation 0 Let r 1le y2 and 6 tan391X x Find the joint and marginal distributions STAT 702J702 BHabing Univ ofSC w 361 Special Case 1 Convolution In general say ZXY We can find a general formula for FZZPZltZ simply by finding the appropriate area under fxy Taking the derivative then gives us the pdf STAT 702J702 BHabing Univ ofSC w Example X and Y are exponential RVs with parameter 9 STAT 702J702 BHabing Univ ofSC w 361 Sgecial Case 2 Quotient A general formula for the quotient ZYX can also be derived by examining the CDF To do this easily note that if yxsz then if xgt0 we have y g xz and if xlt0 then y 2 xz STAT 702J702 BHabing Univ ofSC w w Back to the earlier example X and Y have joint pdf fXYXy 2 Osxltyg1 UXY STAT 702J702 BHabing Univ ofSC w n 37 Order Statistics Let X1 X2 Xn be independent random variables with the same CDF FXX The values in order from lowest to smallest are the order statistics STAT 702J702 BHabing Univ ofSC V4