1. CS723 - Probability and Stochastic Processes

2. Lecture No. 19

3. In Previous Lectures
Analysis of jointly random continuous random variables using their joint PDF
Independence of two random variables thorough separation of their joint PDF
Graphical characteristics of the joint PDF of two independent random variables
Joint failure of two TV sets modeled using separable joint PDF
Probabilities of interesting joint event using integration of joint PDF

4. Exponential RV Pair
The joint PDF is fXY(x,y) = e-(2x+y)/20 fXY(x,y) = fX(x) fY(y) = (0.1e-x/10 )(0.05e-y/20) A = {(x,y) s.t. x+y < 20} B = {(x,y) s.t. x+y > 20 & x > y}

5. Exponential RV Pair

6. Pair of Erlang RV’s

7. Pair of Dependent RV’s
fXY(x,y) defined for (x,y) ε [0,10]x[0,10] fXY(x,y) = K(2x + y) = (2x + y)/1500 fX(x) = (5 + 2x)/ & fY(y) = (10 + y)/150

8. Trajectories on Joint PDF

9. Pair of Dependent RV’s
fXY(x,y) defined for (x,y) ε [0,10]x[0,10] fXY(x,y) = K(4 + x - 0.4y) = (20 + 5x - 2y)/3500 fX(x) = (2 + x)/70 & fY(y) = (45 - 2y)/350

10. Trajectories on Joint PDF

11. Pair of Dependent RV’s
fXY(x,y) defined for (x,y) ε [0,∞)x[0, ∞) fXY(x,y) = K( e-x/10 e-y/20 e-xy/30) Portion of trajectories are shown .

12. Pair of Gaussian RV’s
fXY(x,y) defined for (x,y) ε (-∞, ∞)x(-∞, ∞) fXY(x,y) = K(e-αx2 e-βy2 e-γxy) = K e-(αx2 + βy2 + γxy)

13. Trajectories on Joint PDF

14. Correlation & Covariance
Joint PDF hold the complete information about the behavior of two RV’s
The means and variances are important characteristics of individual RV’s
Correlation is a mixed higher moment that describes relationship between two RV’s
Positive (negative) correlation implies that if one random variable assumes a large value, then the other is more likely to assume a large (small) value
No correlation between independent RV’s

15. Correlation & Covariance
Correlation between X & Y is denoted by RXY
Covariance of X & Y is denoted by CovXY
Correlation Coefficient