Multivariate Probability Distributions Random Vectors and Joint Probability Distributions The Continuous Case Definition 5(5.5).[Joint PDF] Two random variables X and Y are said to have a continuous joint distribution if their joint CDF FxY(,y)is absolutely continuous in both x and y.In this case,there exists a nonnegative function fxy(x,y)such that for any subset A CR2, PX)∈N=∫e fxr(,y)dxdy In particular, Bxve》=fxa,hd The function fxy(x,y)is called a joint PDF of (X,Y). Multivariate Probability Distributions Introduction to Statistics and Econometrics Juy1,2019 26/370
Multivariate Probability Distributions Multivariate Probability Distributions Introduction to Statistics and Econometrics July 1, 2019 26/370 Random Vectors and Joint Probability Distributions The Continuous Case Definition 5 (5.5). [Joint PDF]
Multivariate Probability Distributions Random Vectors and Joint Probability Distributions The Continuous Case Lemma 3(5.4).[Properties of Joint PDF fxy(x,y) (1)fxy(x,y）≥0 for all(x,y）in the xy plane; (2)∫∫efxY(x,y)drdy=1. (x (y) Lx=0 fx(a)=Area (x,y=b) Surfacefyx,y) fy(b)=Area fx.y(x=o,y) Multivariate Probability Distributions Introduction to Statistics and Econometrics Juy1,2019 27/370
Multivariate Probability Distributions Multivariate Probability Distributions Introduction to Statistics and Econometrics July 1, 2019 27/370 Random Vectors and Joint Probability Distributions The Continuous Case Lemma 3 (5.4). [Properties of Joint PDF 𝑓𝑋𝑌(𝑥, 𝑦)]
Multivariate Probability Distributions Random Vectors and Joint Probability Distributions The Continuous Case Proof: ●(1)Denoting A(x,y）={(u,w):u≤c,v≤y}for any given pair(c,y)∈R2, we have P[(X,Y)∈A(x,y)]=P(X≤x,Y≤y) Fxy(x,y) fxr(u,v)dvdu. To be Continued Multivariate Probability Distributions Introduction to Statistics and Econometrics Juy1,2019 28/370
Multivariate Probability Distributions Multivariate Probability Distributions Introduction to Statistics and Econometrics July 1, 2019 28/370 Random Vectors and Joint Probability Distributions The Continuous Case Proof: To be Continued
Multivariate Probability Distributions Random Vectors and Joint Probability Distributions The Continuous Case At the points of (x,y)where Fxy(x,y)is differentiable, we have fxy(x,y)= 82Fxy(x,y) ∂x∂y 20, where the equality follows from the fundamental theorem of calculus,and the inequality follows from the fact that Fxy(x,y)is nondecreasing in (x,y). (2)The integralfxr(,y)dady-1 follows im- mediately from the fact that Fxy(oo,oo)=1. Multivariate Probability Distributions Introduction to Statistics and Econometrics July1,2019 29/370
Multivariate Probability Distributions Multivariate Probability Distributions Introduction to Statistics and Econometrics July 1, 2019 29/370 Random Vectors and Joint Probability Distributions The Continuous Case
Multivariate Probability Distributions Random Vectors and Joint Probability Distributions The Continuous Case Question:What is the interpretation of the joint PDF fxy(x,y)? For any given pair (x,y)in the xy-plane,consider the event A,g》={红-5<X≤t+andg-5<y≤g+}, where e>0 is a small constant. .A(x,y)is the event that (X,Y)takes values in a small rectangular area centered at point (z,y)and with each side equal to c. Multivariate Probability Distributions Introduction to Statistics and Econometrics Juy1,2019 30/370
Multivariate Probability Distributions Multivariate Probability Distributions Introduction to Statistics and Econometrics July 1, 2019 30/370 Random Vectors and Joint Probability Distributions The Continuous Case