1、Joint and Conditional Distribution I9th week / Probability and Statistics (III)Objectives of This WeekUnderstand the relation between two random variables given by a joint probability distributionUnderstand the conditional probability distributionLearn the concepts of independency, covariance, and c
2、orrelationPractice various probability relations with RJoint Probability Distribution Joint probability r, describes the relation of two random variable and Joint CDF, , = , 3X , Y y(, y)XYProperties of Joint CDF1. 0 , , 12. When 1 2 and 1 2, , 1 ,1 ,(2,2)3. , , = Pr , = (); marginal cdf4. , , =1, ,
3、 , =0, , , =?, , , =?5. Pr 1 2,1 2 = , 2 ,2 , 1 ,2, 2 ,1 +, 1,14Discrete Random Variables Continuous Random Variables , , = Pr = , = , , = Pr +, + / = , , = , , = , , = , , , , = 1 , , =1, , = , , , , = , , , , =2, , /rA = (,), , rA=, , Joint PMF and PDF5Joint PMF and PDF Example: , , = for 0 2 and
4、0 1, what is ? what is Pr + 1 ?6Covariance Covariance One simple value to represent the relation between two random variables Represent how two random variables vary together Uncorrelated: When , = 0, = = , = , 7Covariance Properties of Covariancea, = , = Correlation coefficient Covariance scaled by variance Strictly between -1 and 1, = , 8Covariance Correlation coefficient is scale-free, but covariance is notHigh Cov, High R1 111Low Cov, Low R1 111High Cov (neg), High R (neg)1 111Low Cov, High R0.1 0.10.10.19
