1. CS723 - Probability and Stochastic Processes

2. Lecture No. 28

3. 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

4. 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

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

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

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

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

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

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

11. KUV from KXY

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

13. 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
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