1. CS723 - Probability and Stochastic Processes

2. Lecture No. 22

3. In Previous Lectures
A scalar-valued random variable that is a function of two RV’s Examples: Z = X+Y and Z = Y – X
The example involved two televisions, the sum of their service times, and difference of their service times
PDF and CDF of Z from the joint PDF of X & Y Answer questions about Z directly by processing fZ(z)

4. Function of Two RV’s
Ratio of service times of two TV’s Z = Y / X

5. Evaluation of FZ(z)

6. CDF of Z = Y / X
FZ(z) = z / (z + 2)

7. PDF of Z = Y / X
fZ(z) = 1/(2+z) – z/(2+z)2

8. Events for Z = Y / X
Pr( 0.5 < Z ) & Pr( Z > 2 )

9. Function of Two RV’s
Multiplication of service times Z = XY

10. Evaluation of FZ(z)

11. Evaluation of FZ(z)
The double integral to get FZ(z) is given by

12. Function of Two RV’s
Max of service times of two TV’s Z = max(X,Y)

13. Evaluation of FZ(z)
The parameter z only takes on positive values The area of integration is always a square Due to independence, the probability of this square is easy to find

14. CDF of Z = max(X,Y)

15. PDF of Z = max(X,Y)

16. Events for Z = max(X,Y)
Pr( Z < 3 ) & Pr( Z > 30)

17. Function of Two RV’s
Min of service times of two TV’s Z = min(X,Y)

18. Evaluation of FZ(z)
The parameter z only takes on positive values The area of integration is always a square Due to independence, the probability of this square is easy to find

19. CDF of Z = min(X,Y)

20. Function of Two RV’s
Distance from origin to (x,y) Z = (X2 + Y2)1/2