CS723 - Probability and Stochastic Processes

Lecture No. 28

In Previous Lectures Finished transformation of random variables Started discussion of mean and variance of transformed random variables fUV(u,v) can be used to find expected values of functions of U, V, or both fXV(x,y) can be used to find expected values of functions of X, Y, or both fXV(x,y) can be directly used to find expected values of functions of U, V, or both if transforming functions u=g( . , . ) and h( . , .) are known

E[r( u, v)] using fXY( x, y) Expected value of any function of U and V can be found using joint PDF fXY(x,y) and the transforming functions involved

Linear g( . , . ) and h( . , . )

Linear g( . , . ) and h( . , . )

Linear g( . , . ) and h( . , . ) Correlation and Covariance of U and V are:

Covariance Matrices Covariance matrix for uncorrelated X and Y is Then, covariance matrix for U and V is

Covariance Matrix Covariance matrix for U & V can be written as Is there a relationship between Covariance matrices KUV and KXY

Covariance Matrix Covariance matrix for U & V can be written as Uncorrelated X & Y can give correlated U & V

KUV from KXY

KUV from KXY For correlated X and Y Is it possible for U and V to be uncorrelated when X and Y are correlated

Eigen Values & Eigenvectors If Ta = λa, then λ is called an eigen value of T and a is called an eigenvector of T For non-trivial 2x2 matrices, there are two eigen values λ1, λ2 associated with two corresponding eigenvectors a1, a2 Then, Ta1 = λ1a2 and Ta2 = λ2a2 Check before uploaded it’s part of lect 28 or 29