Mathematical ExpectationExpected ValueSlide 3Law of the Unconscious StatisticianSlide 5VarianceSlide 7Bivariate DistributionsDiscrete Bivariate DistributionContinuous Bivariate DistributionSlide 11Covariance and CorrelationSlide 13Slide 14Slide 15Linear CombinationsSlide 17Chebyshev’s InequalityBell-shaped DistributionsComparisonChebyshev’s Inequality ExamplesSlide 22Slide 231Mathematical ExpectationFor a r.v. X or a function g(X)•Mean or Expected Value•VarianceFor two r.v.’s X and Y or a function g(X, Y)•Expected Value•Covariance and CorrelationLinear Combinations of r.v.’sChebyshev’s Inequality2Expected ValueFor a discrete r.v. X with p.m.f. fX(x) For a continuous r.v. X with p.d.f. fX(x)Some basic rules: a, b are nonrandom•E[a] = a•E[aX] = a E[X] •E[aX + b] = a E[X] + b dxxfxXEXX)(][xXXxfxXE )(][3Expected ValueDiscrete Example: Flip 2 coins fX(0) = 1/4, fX(1) = 1/2, fX(2) = 1/4E[X] = (0)(1/4) + (1)(1/2) + (2)(1/4) = 1 on average we expect to see one headE[4X+3] = 4 E[X] + 3 = 4(1) + 3 = 1Continuous Example: fX(x) = 2(x1) for 1 < x < 2; o/w fX(x) = 0352)1(2][21221dxxxdxxxXE4Law of the Unconscious StatisticianFor a discrete r.v. X with p.m.f. fX(x), the expected value for a function g(X) isFor a continuous r.v. X with p.d.f. fX(x), the expected value for a function g(X) isThe distribution for g(X) is not needed.xXXgxfxgXgE )()()]([)( dxxfxgXgEXXg)()()]([)(5Law of the Unconscious StatisticianDiscrete Example: Flip 2 coins fX(0) = 1/4, fX(1) = 1/2, fX(2) = 1/4Continuous Example: fX(x) = 2(x1) for 1 < x < 2; o/w fX(x) = 0 6172)1(2][21232122dxxxdxxxXE4741123211134110313XE6VarianceFor any r.v. X, the variance is the expected squared deviation about its mean:•Note that variance cannot be negativeLaw of the Unconscious StatisticianSome basic rules: a, b are nonrandom•V(a) = 0•V(aX) = a2 V(X) •V(aX + b) = a2 V(X) 2222][])[()(XXXXEXEXV]))([())((2)(2)( XgXgXgEXgV7VarianceDiscrete Example: Flip 2 coins fX(0) = 1/4, fX(1) = 1/2, fX(2) = 1/4 ; E[X] = 1E[X 2] = (02)(1/4) + (12)(1/2) + (22)(1/4) = 3/2 V(X) = E[X 2]  (E[X])2 = 3/2  (1)2 = 1/2 V(4X+3) = (4)2V(X) = (16)(1/2) = 8Continuous Example:18135617])[(][)(222 XEXEXV617][;35][2 XEXE8Bivariate DistributionsFor a discrete r.v.’s X and Y with joint p.m.f. fX,Y (x, y), the expected value for a function g(X,Y) isFor a continuous r.v.’s X and Y with joint p.d.f. fX,Y (x, y), the expected value for a function g(X,Y) isx yYXYXgyxfyxgYXgE ),(),()],([,),(  dxdyyxfyxgYXgEYXYXg),(),()],([,),(9Discrete Bivariate DistributionExample: X Y 0 1 2 5 0.2 0.1 0.1 10 0.1 0.2 0.3 14.0)3.0(102)2.0(101)1.0(100)1.0(52)1.0(51)2.0(50YXE 5.9)3.0)(10)(2()2.0)(10)(1()1.0)(10)(0()1.0)(5)(2()1.0)(5)(1()2.0)(5)(0(XYE10Continuous Bivariate DistributionExample: for 0 < x < 1, 0 < y < 1)2(512),(,yxxyxfYX 101042310102,22])2([512)2()(512),()(][dydxyxyyxdydxyxxyxdxdyyxfyxYXEYX11Continuous Bivariate DistributionExample (cont’d):25451259353225184110351254)2(512][103210210105422yydyyydyxyxyyYXE12Covariance and CorrelationCovariance measures the association between two r.v.’s X and Y Correlation is the scale-free version of covariance YXYXXYXYEYXEYX][)])([(),(Cov11)()(),(CovXYYXXYXYYVXVYX13Covariance between two r.v.’s X and Y:•Cov(X,Y) > 0  Positive relationship.•Cov(X,Y) < 0  Negative relationship.•Cov(X,Y) = 0  No relationship.Correlation between two r.v.’s X and Y:•ρXY close to 1  Strong positive correlation.•ρXY close to 1  Strong negative correlation.•ρXY = 0  No correlation.Covariance and Correlation14Covariance and CorrelationDiscrete Example: X Y 0 1 2 fY (y) 5 0.2 0.1 0.1 0.4 10 0.1 0.2 0.3 0.6 fX (x) 0.3 0.3 0.4 1.0  5.9XYE0.70)6.0)(10()4.0)(5(][0.8)6.0)(10()4.0)(5(][9.1)4.0)(2()3.0)(1()3.0)(0(][1.1)4.0)(2()3.0)(1()3.0)(0(][2222222YEYEXEXE15Covariance and CorrelationDiscrete Example (cont’d):X and Y are somewhat positively correlated.7.0)0.8)(1.1(5.9])[])([(][),(Cov YEXEXYEYX0.6)8(70])[(][)(69.0)1.1(9.1])[(][)(222222YEYEYVXEXEXV344.0)0.6()69.0(7.0)()(),(CovYVXVYXXY16Linear CombinationsX and Y are r.v.’s•E[aX + bY + c] = a E[X] + b E[Y] + c•E[a g(X) + b h(Y) + c] = a E[g(X)] + b E[h(Y)] + c•V(aX + bY) = a2 V(X) + b2 V(Y) + 2ab Cov(X, Y)Discrete Example (cont’d):•E[4X  3Y + 2] = (4)(1.1)  (3)(8) + 2 = 17.6•V(4X  3Y) = (4)2(0.69) + (3)2(6) + 2(4)(3)(0.7) = (16)(0.69) + (9)(6)  (24)(0.7) = 48.240.6)(;0.8][;69.0)(;1.1][  YVYEXVXE7.0),(Cov YX17Linear CombinationsX and Y are independent r.v.’s•E[XY] = E[X] E[Y] •Cov(X, Y) = E[XY]  (E[X])(E[Y]) = 0•V(aX + bY) = a2 V(X) + b2 V(Y)X1, X2, … , Xn are mutually independent r.v.’s• niiinnnnXVaXVaXVaXVaXaXaXaV1222221212211)()()()()(18Chebyshev’s InequalityFor any distribution, the probability that r.v. X will take on a value within   k is211][kkXkP 19Bell-shaped DistributionsFor bell-shaped symmetric distributions•Approximately 68% of values within   •Approximately 95% of values within   2•Nearly 100% of values within   320Comparison21Chebyshev’s Inequality ExamplesA r.v. X has mean 8 and variance 16, but its distribution is unknown.Approximately 67.3% of the possible values of X lie between 1 and