CS723 - Probability and Stochastic Processes

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CS723 - Probability and Stochastic Processes

Lecture No. 19

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

Exponential RV Pair The joint PDF is fXY(x,y) = 0.005 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}

Exponential RV Pair

Pair of Erlang RV’s

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)/150 & fY(y) = (10 + y)/150

Trajectories on Joint PDF

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

Trajectories on Joint PDF

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 .

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)

Trajectories on Joint PDF

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

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