CS723 - Probability and Stochastic Processes

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CS723 - Probability and Stochastic Processes

Lecture No. 20

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

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]

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

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

Correlation & Covariance fXY(x,y) = {2(x - 55/9) + (y – 50/9)}/1500 for x ε [-55/9,35/9] and y ε [-50/9,40/9] X = 0 , Y = 0 , RXY = CovXY = -50/81

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

Correlation & Covariance fXY(x,y) = (5x - 2y + 20)/3500 X = 130/21, Y = 95/21, CovXY = 250/441

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

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

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

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

Correlated Gaussian RV’s

Correlated Gaussian RV’s