1. CS723 - Probability and Stochastic Processes

2. Lecture No. 27

3. In Previous Lectures Transformation of random variables
Derived PDF of random variable (s) from the PDF of underlying random variable (s) Examples: Y = g(X) Z = g(X,Y) U = g(X,Y) & V = h(X,Y)
fY(y) / fUV(u,v) were found via CDF and via direct processing of fX(x) / fXY(x,y)
fZ(z) was obtained from FZ(z) that was computed from fXY(x,y)

4. fY(y) Directly from fX(x)

5. fY(y) Directly from fX(x)

6. fuv(U,V) from fXy(x,y)

7. fuv(U,V) from fXy(x,y)

8. fZ(z) from fXy(x,y)

9. E[Y=g(X)] from fX(x)

10. E[Y=g (X)] from fX(x)

11. E[Y=g(X)] from fX(x)
Some examples of simple transformations

12. Moments from fXY(x,y) Direct use of fXY(x,y), the joint PDF of X and Y
RXY and ρXY HAVE to use joint PDF fXY(x,y)

13. Moments of U&V from fXY(x,y)
E[g(x,y)] using joint PDF fXY(x,y)

14. Moments of U&V from fXY(x,y)
E[g(x,y)] using joint PDF fXY(x,y)

15. Moments of U&V from fXY(x,y)

16. Moments of U&V from fXY(x,y)
Similarly,
Examples of linear transformations functions g( . , . ) and h( . , . )
To be continued …