IE3301 Fall 2013Chapter 3Chapter 3 cont’d...Slide 4Slide 5Slide 6Chapter 4Chapter 4 cont’d...Slide 9Slide 10Homework AssignmentsIE3301 Fall 2013IE3301 Fall 2013Overview of Chapter 3: Random Variables Overview of Chapter 3: Random Variables and Probability Distributionsand Probability DistributionsOverview of Chapter 4: Mathematical Overview of Chapter 4: Mathematical ExpectationExpectationHomework AssignmentsHomework AssignmentsRandom variable (r.v.), denoted by Random variable (r.v.), denoted by XX•Discrete vs. ContinuousDiscrete vs. Continuous•Each simple event in Each simple event in SS maps to a real number maps to a real number xx•Events: [Events: [X X = = xx], [], [X X < < xx], [], [X X  xx], [], [aa  X X  bb]]ExamplesExamples•Flip a coin: Flip a coin: X X = # of heads= # of heads•Light bulb: Light bulb: X X = lifetime of the bulb in hours= lifetime of the bulb in hoursChapter 3Chapter 3Probability distributionProbability distribution•Discrete: Discrete: Probability mass functionProbability mass function (p.m.f.) (p.m.f.) ffXX((xx) = ) = PP[[X X = = xx]]•Continuous: Continuous: Probability density functionProbability density function (p.d.f.) (p.d.f.) ffXX((xx) is NOT a probability) is NOT a probability•Cumulative distribution functionCumulative distribution function (c.d.f.) (c.d.f.) FFXX((xx) = ) = PP[[X X  xx] ] Chapter 3 Chapter 3 cont’d...cont’d...Joint probability distributionsJoint probability distributions•More than one r.v.More than one r.v.Bivariate Distributions (Bivariate Distributions (XX and and YY) ) •Discrete joint p.m.f : Discrete joint p.m.f : ffXX,,YY((xx,, y y) = ) = PP[([(X X = = xx))((Y Y = = yy)] = )] = PP[[X X = = xx, , Y Y = = yy]]•Continuous joint p.d.f : Continuous joint p.d.f : ffXX,,YY((xx,, y y) ) Chapter 3 Chapter 3 cont’d...cont’d...Chapter 3 Chapter 3 cont’d...cont’d...Marginal distributions (bivariate case)Marginal distributions (bivariate case)•for for XX : : ffXX((xx))•for for YY : : ffYY((yy))Conditional distributions for Conditional distributions for XX and and YY•Distribution of Distribution of XX given [ given [Y Y = = yy]:]: f fXX||YY((x x | | yy) = ) = ffXX,,YY((xx,, y y) / ) / ffYY((yy) for ) for ffYY((yy) > 0) > 0•Distribution of Distribution of YY given [ given [X X = = xx]:]: f fYY||XX((y y | | xx) = ) = ffXX,,YY((xx,, y y) / ) / ffXX((xx) for ) for ffXX((xx) > 0) > 0Chapter 3 Chapter 3 cont’d...cont’d...IndependenceIndependence•Recall rules for independent eventsRecall rules for independent eventsXX and and YY are independent r.v.’s if and only if are independent r.v.’s if and only if•ffXX||YY((x x | | yy) = ) = ffXX((xx) or) or•ffYY||XX((y y | | xx) = ) = ffYY((yy) or) or•ffXX,,YY((xx,, y y) = ) = ffXX((xx) ) ffYY((yy) )Chapter 4Chapter 4Mean or Expected Value of a r.v. Mean or Expected Value of a r.v. XXXX = E = E[[XX]]•Weighted average of values of Weighted average of values of XX, where the , where the weights are the probabilities.weights are the probabilities.Expected Value of a function Expected Value of a function gg((XX))gg((XX)) = E = E[[gg((XX)])]•Law of the Unconscious StatisticianLaw of the Unconscious Statistician•Do not need the distribution of Do not need the distribution of gg((XX))Chapter 4 Chapter 4 cont’d...cont’d...Variance of a r.v. Variance of a r.v. XXXX2 2 = V= V((XX) or Var() or Var(XX) = ) = EE[([(X X  XX))22]]•Average squared deviation about the mean.Average squared deviation about the mean.Variance of a function Variance of a function gg((XX))gg((XX) ) 22 = = Var(Var(gg((XX)) = )) = EE[([(gg((XX))  gg((XX)) ))22]]•Also uses Law of the Unconscious StatisticianAlso uses Law of the Unconscious StatisticianChapter 4 Chapter 4 cont’d...cont’d...Covariance between r.v.’s Covariance between r.v.’s XX and and YYXYXY = = Cov(Cov(XX, , YY) = ) = EE[([(X X  XX)()(Y Y  YY)])]Correlation between r.v.’s Correlation between r.v.’s XX and and YYXYXY = = XYXY // ( (XX YY ) )Linear Combination of r.v.’sLinear Combination of r.v.’s•aXaX + + bb•aXaX + + bYbY, , gg((XX) + ) + hh((YY))•aa11XX11 + … + + … + aannXXnnChapter 4 Chapter 4 cont’d...cont’d...Chebyshev’s InequalityChebyshev’s Inequality•Valid for all distributionsValid for all distributions•Conservative for bell-shaped distributionsConservative for bell-shaped distributionsHomework AssignmentsHomework AssignmentsHW #3 due Wednesday, September 18HW #3 due Wednesday, September 18•Pg 91: 3.1, 3.3, 3.5a, 3.7, 3.13, 3.15Pg 91: 3.1, 3.3, 3.5a, 3.7, 3.13, 3.15•pg 104: 3.37, 3.43a, 3.49, 3.54 pg 104: 3.37, 3.43a, 3.49, 3.54 HW #4 due Wednesday, September 25HW #4 due Wednesday, September 25•pg 117: 4.1, 4.7, 4.9, 4.13pg 117: 4.1, 4.7, 4.9, 4.13•pg 127: 4.35, 4.39, 4.45pg 127: 4.35, 4.39, 4.45•pg 137: 4.53, 4.57, 4.77, 4.65pg 137: 4.53, 4.57, 4.77,