Bivariate Continuous Random Variables.Summary using a single bivariate density function.Juana [email protected] Department of StatisticsJ. Sanchez Joint densities: Bivariate ContinuousIn this lectureSupplementary recommended reading in Ross: Selections from Theory inChapter 6 and chapter 7. Will go over all the important theory onbivariate continuous random variables that you need to know using oneexample. More examples after this lecture.I. Marginal distributions, expectations and variances. Independence.II. Expectation of the product of two random variablesIII. Covariance and correlationIV. Expectations of linear functions of two random variables.V. Variance of linear functions of two random variables.VI. Conditional distributions, expectations and variances.Note that for all the above densities you need to know how to computeprobabilities with them.J. Sanchez Joint densities: Bivariate ContinuousJoint Bivariate example used throughout this lecturef (x1, x2) =3x10 ≤ x2≤ x1≤ 10 otherwise0.00.51.00.00.51.00.00.51.01.52.0J. Sanchez Joint densities: Bivariate ContinuousI. Marginal distributions, expectations and variances for theexample.f (x1) =Rx103x1dx2= 3x210 ≤ x1≤ 1µx1= E (X1) =R10x1f (x1)dx1=R10x1(3x21)dx1= 3/4σ2x1=R10(x1− µx1)2f (x1)dx1= E (x21) −[E (x1)]2= (3/5) − (3/4)2= 0.0375f (x2) =R1x23x1dx1=32(1 −x22) 0 ≤ x2≤ 1µx2= E (X2) =R10x2f (x2)dx2=R10x2(3/2)(1 −x22)dx2= 3/8σ2x2=R10(x2− µx2)2f (x2)dx2= E(x22)−[E(x2)]2= 1/5 − (3/8)2= 0.0594J. Sanchez Joint densities: Bivariate ContinuousIndep endence of two random variablesIndependence of random variablesTwo random variables X1and X2are independent iff (x1, x2) = f (x1)f (x2)In the example, f (x1) = 3x21and f (x2) =32(1 −x22)f (x1)f (x2) = 3x2132(1 −x22) =92x21− x21x22Butf (x1, x2) = 3x1So because f (x1)f (x2) 6= f (x1, x2), X1and X2are not independent. Andthis implies that f (x1| x2) 6= f (x1) and f (x2| x1) 6= f (x2).J. Sanchez Joint densities: Bivariate ContinuousI I. Exp ectation of the pro duct of two random variables inthis exampleE (x1x2) =R10Rx10(x1x2) 3x1dx2dx1= 3R10Rx10x21x2dx2dx1= 3R10x21hx222ix10dx1=32R10x41dx1=32hx515i10=310J. Sanchez Joint densities: Bivariate ContinuousI II. Covariance and correlation for the exampleCov (X1, X2) = E [(X1− 3/4)(X2− 3/8)]=Z10Zx10[(X1− 3/4)(X2− 3/8)] 3x1dx2dx1= E (X1X2) −E (X1)E (X2)= 3/10 −(3/4)(3/8)= 0.0175ρ =Cov (X1, x2)σx1σx2=0.0175√0.0375√0.0594= 0.39J. Sanchez Joint densities: Bivariate ContinuousCorrelationρ is a measure of the strength and direction of a linear associationbetween two random variables.ρ is a number between -1 and 1.Small ρ or ρ close to 0 could mean that there is no linear association,but there could be nonlinear association. Scatter plot of the base ofthe distribution of the two variables is needed to determine that.J. Sanchez Joint densities: Bivariate ContinuousIV. Expectation of linear functions of the two randomvariables in the exampleLet the linear function be Y = X1− X2.E (X1− X2) =Z10Zx10(x1− x2)3x1dx2dx1=Z10Zx10x1(3x1)dx2dx1−Z10Zx10x2(3x1)dx2dx1=Z10x1Zx10(3x1)dx2dx1+Z10x2Z1x2(3x1)dx1dx2=Z10x1f (x1)dx1+Z10x2f (x2)dx2= E (X1) − E (X2)= 3/4 − 3/8= 3/8J. Sanchez Joint densities: Bivariate ContinuousLet a1, a2be constant numbers.Expectation of a linear function of two random variablesE (a1X1+ a2X2) = a1E (X1) + a2E (X2)In the example we solved, a1= 1; a2= −1.For the joint bivariate density example, seen in this lecture,E (2X1+4X2) = 2E (X1)+4E (X2) = 2(3/4)+4(3/8) = (6/4)+(12/8) = 3J. Sanchez Joint densities: Bivariate ContinuousV. Variance of the linear combination of two randomvariablesVar (X1− X2) =Z10Zx10[(x1− x2) − (µx1− µx2)]23x1dx2dx1=Z10Zx10(x1− µx1)23x1dx2dx1+Z10Zx10(x2− µx2)23x1dx2dx1−2Z10Zx10[((x1− µx1)(x2− µx2))] 3x1dx2dx1=Z10(x1− µx1)2Zx103x1dx2dx1+Z10(x2− µx2)2Z1x23x1dx1dx2−2Z10Zx10[((x1− µx1)(x2− µx2))] 3x1dx2dx1= Var (X1) + Var(X2) − 2Cov (X1, X2)= 0.0375 + 0.0594 − 2(0.0175)= 0.0619J. Sanchez Joint densities: Bivariate ContinuousLet a1, a2be constant numbers.Variance of linear functions of two random variablesVar (a1X1+ a2X2) = a21Var (X1) + a22Var (X2) + 2(a1a2)Cov (X1, X2)J. Sanchez Joint densities: Bivariate ContinuousVI. Conditional Densities and conditional expectations andvariancesf (x2| x1) =f (x1, x2)f (x1)=3x13x21=1x10 ≤ x2≤ x1For example, if x1= 0.5,f (x2| x1= 0.5) =10.5= 2 0 ≤ x2≤ 0.5E (X2| X1= 0.5) =Z0.50x2(2)dx2= 1/4Var (X2| X1= 0.5) =Z0.50(x2− 1/4)22dx2= E (X2| X1= 0.5) − (1/4)2=Z0.50x222dx 2− (1/16)= 1/12 −1/16= 0.0208J. Sanchez Joint densities: Bivariate ContinuousSimilarly, we can find the conditional distributionf (x1| x2) =f (x1, x2)f (x2)=3x1(3/2)(1 −x22)x2≤ x1≤ 1For example, if x2= 0.5,f (x1| x2= 0.5) =3x1(3/2)(1 −(1/2)2)= (8/3)x10.5 ≤ x1≤ 1E (X1| X2= 0.5) =Z10.5x1(8/3)x1dx1= 4/3andVar (X1| X2= 0.5) =Z10.5x21(8/3)x1dx1− (4/3)2J. Sanchez Joint densities: Bivariate ContinuousVI I. ProbabilitiesA question on probability will have to be answered with the right density.P(X1> a): use the marginal density for X1, f (x1).P(a < X2< b): use the marginal density for X2, f (x2).P(X1< a | X2= b): Use the corresponding conditional densityfunction of X1, f (xi| x2= b)P(X1+ X2< a): Use the joint density, f (x1, x2).J. Sanchez Joint densities: Bivariate ContinuousPrepareStudy the example in this lecture.Study the theory and examples seen in the complementary largerlecture notes postedJ. Sanchez Joint densities: Bivariate
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