Multivariate Probability Distributions Random Vectors and Joint Probability Distributions Random Vectors and Joint Probability Distributions Remarks: Suppose Fx()and Fy()are strictly increasing functions. Then Fxy(x,y） =P(X≤x,Y≤y) P[Fx(X)≤Fx(x),Fy(Y)≤Fy(y)] P[U≤Fx(x),V≤Fy(y)] CFx(x),Fy(y). Multivariate Probability Distributions Introduction to Statistics and Econometrics Juy1,2019 11/370
Multivariate Probability Distributions Multivariate Probability Distributions Introduction to Statistics and Econometrics July 1, 2019 11/370 Random Vectors and Joint Probability Distributions Remarks: Random Vectors and Joint Probability Distributions
Multivariate Probability Distributions Random Vectors and Joint Probability Distributions Random Vectors and Joint Probability Distributions This suggests that the joint distribution Fxy(x,y)can be de- composed into two separate components:the marginal distribu- tions Fx()and Fy()on one hand,and the "pure"dependence function C(,)between X and Y on the other hand. C(,)only specifies the dependence or association between X and Y regardless of their marginal distributions.It separates the marginal behaviors from the association between X and Y. Given the functional form C(,),different marginal distribu- tions will yield different joint distributions of (X,Y). The copula has been widely used in financial econometrics and financial industries,to model comovements among markets or among assets. Multivariate Probability Distributions Introduction to Statistics and Econometrics July1,2019 12/370
Multivariate Probability Distributions Multivariate Probability Distributions Introduction to Statistics and Econometrics July 1, 2019 12/370 Random Vectors and Joint Probability Distributions Random Vectors and Joint Probability Distributions
Multivariate Probability Distributions Random Vectors and Joint Probability Distributions The Discrete Case Definition 3(5.3).[Joint PMF] Let X and Y be two DRV's.Then their joint PMF is defined as fxy(x,y)=P(X=x nY=y)=P(X=x,Y=y) for any point (x,y)ER2. RXX.Y X1 X2 P(Y 0.1 0.2 0.3 Y2 0.3 0.4 0.7 Px(X) 0.4 0.6 X2 X1 Multivariate Probability Distributions Introduction to Statistics and Econometrics July1,2019 13/370
Multivariate Probability Distributions Multivariate Probability Distributions Introduction to Statistics and Econometrics July 1, 2019 13/370 Random Vectors and Joint Probability Distributions The Discrete Case Definition 3 (5.3). [Joint PMF]
Multivariate Probability Distributions Random Vectors and Joint Probability Distributions The Discrete Case Lemma 2 (5.3).[Properties of fxy(x,y)] (1)fxy(x,y)≥0 for all(x,y)∈R2. (2)∑m∈nx∑weay fxr(c,)=l,where x and y are the supports of X and Y respectively. Multivariate Probability Distributions Introduction to Statistics and Econometrics Juy1,2019 14/370
Multivariate Probability Distributions Multivariate Probability Distributions Introduction to Statistics and Econometrics July 1, 2019 14/370 Random Vectors and Joint Probability Distributions The Discrete Case Lemma 2 (5.3). [Properties of 𝑓𝑋𝑌(𝑥, 𝑦)]
Multivariate Probability Distributions Random Vectors and Joint Probability Distributions The Discrete Case Definition 4 (5.4).[Support] The support of a bivariate random vector (X,Y)is defined as the set of all possible pairs of (x,y)which (X,Y)will take with strictly positive probability.That is, Support(X,Y)=xy ={(x,y)ER2:fxy(,y)>0. Multivariate Probability Distributions Introduction to Statistics and Econometrics July1,2019 15/370
Multivariate Probability Distributions Multivariate Probability Distributions Introduction to Statistics and Econometrics July 1, 2019 15/370 Random Vectors and Joint Probability Distributions The Discrete Case Definition 4 (5.4). [Support]