1. CS723 - Probability and Stochastic Processes

2. Lecture No. 20

3. In Previous Lectures
Analysis of a pair of continuous random variables using their joint PDF
Independence of two random variables thorough separation of their joint PDF
Correlation, covariance, and correlation coefficient of a pair of random variables measure influence of one random variable on the other.
Correlation coefficient only meaningful for a pair of random variables that are not independent of each other

4. Correlation & Covariance
Correlation between X & Y is denoted by RXY
Covariance of X & Y is denoted by CovXY
Correlation Coefficient
Correlation coefficient always lies in [-1,1]

5. Correlation & Covariance
fXY(x,y) = (2x + y)/1500 for (x,y) ε [0,10]x[0,10] fX(x) = (2x + 5)/ & fY(y) = (y + 10)/150 X = 55/9 , Y = 50/9 , RXY = 100/3

6. Correlation & Covariance
fXY(x,y) = {2(x-2)+(y-3)}/1500 (x,y) ε [2,12]x[3,13] fX(x) = (2x + 1)/ & fY(y) = (y + 7)/150 X = 73/9 , Y = 77/9 , RXY = 619/9

7. Correlation & Covariance
fXY(x,y) = {2(x - 55/9) + (y – 50/9)}/ for x ε [-55/9,35/9] and y ε [-50/9,40/9] X = 0 , Y = 0 , RXY = CovXY = -50/81

8. Correlation & Covariance
fXY(x,y) = (2x + y)/1500 for (x,y) ε [0,10]x[0,10] X = 55/9, Y = 50/9, RXY = 100/3, CovXY = -50/81

9. Correlation & Covariance
fXY(x,y) = (5x - 2y + 20)/ X = 130/21, Y = 95/21, CovXY = 250/441

10. Correlation & Covariance
X = Y = 5, RXY = 25, and CovXY = 0

11. Correlation & Covariance
fXY(x,y) = 1/18 if (x+1) > y > (x-1) & (9 – x) > y > (-9-x) X = Y = 0, RXY = CovXY = 20/3

12. Correlation & Covariance
fXY(x,y) = 1/18 if (x+9) > y > (x-9) & (1 – x) > y > (-1-x) X = Y = 0, RXY = CovXY = -20/3

13. Correlated Gaussian RV’s
Pair of Gaussian random variables have a joint PDF fXY(x,y) that is defined for (x,y) ε (-∞, ∞)x(-∞, ∞) fXY(x,y) = K(e-αx2 e-βy2 e-γxy) = K e-(αx2 + βy2 + γxy) We can try to find the means, variances, correlation coefficient from this PDF but a standard format is usually better

14. Correlated Gaussian RV’s

15. Correlated Gaussian RV’s