Multivariate Probability Distributions Random Vectors and Joint Probability Distributions The Continuous Case Suppose fxy(a,y)is continuous at the point (x,y).Then ry+e/2 x+e/2 fxy(u,v)dudv fxy(元，)·e·e for some(,☑) 义 fxy(x,y)e2 when e is small. Although fxy(x,y)is not a probability measure,it is proportional to the probability that (X,Y)takes values in a small rectangular area centered at point (x,y). Multivariate Probability Distributions Introduction to Statistics and Econometrics July1,2019 31/370
Multivariate Probability Distributions Multivariate Probability Distributions Introduction to Statistics and Econometrics July 1, 2019 31/370 Random Vectors and Joint Probability Distributions The Continuous Case
Multivariate Probability Distributions Random Vectors and Joint Probability Distributions The Continuous Case 。In general.. the probability P(X,Y)E A has a 3- dimensional geometric interpretation. Recall that in the univariate case,when A is an interval on the real line(eg,A={x∈R:a<x≤b}),the probability P(X E A)is equal to the area under the curve fx (x)over the interval A. For the bivariate case,suppose A is an area on the ay plane.Then the probability P(X,Y)E A is a volume under the surface of fxy(x,y)over the area A in the xy plane,as shown in Figure 5.1(b). Multivariate Probability Distributions Introduction to Statistics and Econometrics July1,2019 32/370
Multivariate Probability Distributions Multivariate Probability Distributions Introduction to Statistics and Econometrics July 1, 2019 32/370 Random Vectors and Joint Probability Distributions The Continuous Case
Multivariate Probability Distributions Random Vectors and Joint Probability Distributions The Continuous Case .The geometric interpretation for P(X,Y)EA has im- portant implications: (1)an event that (X,Y)takes a value at any indi- vidual point (x,y),or values at any finite number of points in the xy plane,has probability zero; (2)an event that (X,Y)takes values on any one- dimensional curve in the xy plane has probability zero. Multivariate Probability Distributions Introduction to Statistics and Econometrics Juy1,2019 33/370
Multivariate Probability Distributions Multivariate Probability Distributions Introduction to Statistics and Econometrics July 1, 2019 33/370 Random Vectors and Joint Probability Distributions The Continuous Case
Multivariate Probability Distributions Random Vectors and Joint Probability Distributions The Continuous Case Because Fxy(x,y)and fxy(x,y)can be recovered from each other,they are equivalent in the sense that they contain the same information about the joint distribu- tion of (X,Y).However,it is often more convenient to use fxy(x,y)in practice.Also,like the univariate case, for each joint CDF Fxy(a,y),there may exist some de- gree of arbitrariness in defining the joint PDF fxy(c,y) over a countable set of points (y)on the xy-plane, which will not alter the joint CDF Fxy(x,y)in any way. Multivariate Probability Distributions Introduction to Statistics and Econometrics July1,2019 34/370
Multivariate Probability Distributions Multivariate Probability Distributions Introduction to Statistics and Econometrics July 1, 2019 34/370 Random Vectors and Joint Probability Distributions The Continuous Case
Multivariate Probability Distributions Random Vectors and Joint Probability Distributions The Continuous Case Definition 6 (5.6).[Support of (X,Y)] The support of the bivariate continuous random vector (X,Y) is defined as Support(X,Y)={(x,y）∈R2:fxy(c,y)>0}. Multivariate Probability Distributions Introduction to Statistics and Econometrics Juy1,2019 35/370
Multivariate Probability Distributions Multivariate Probability Distributions Introduction to Statistics and Econometrics July 1, 2019 35/370 Random Vectors and Joint Probability Distributions The Continuous Case Definition 6 (5.6). [Support of (𝑋, 𝑌)]