Multivariate Probability Distributions Bivariate Normal Distribution pdf and cdf 0.4 1 0.3 0.8 0.6 0.2 0.4 0.1 0.2 0 1 2 Multivariate Probability Distributions Introduction to Statistics and Econometrics Juy1,2019 6/370
Multivariate Probability Distributions Multivariate Probability Distributions Introduction to Statistics and Econometrics July 1, 2019 6/370 Bivariate Normal Distribution pdf and cdf
Multivariate Probability Distributions Random Vectors and Joint Probability Distributions Random Vectors and Joint Probability Distributions Lemma 1(5.1).[Properties of Fxy(x,y)] (1)Fxy(-o0,y)=Fxy(x,-0o)=0;Fxy(oo,o)=1. (2)Fxy(x,y)is non-decreasing in both x and y. (3)Fxy(a,y)is right continuous in both x and y. 0.8 0.6 0.4 0.2 Multivariate Probability Distributions Introduction to Statistics and Econometrics Juy1,2019 71370
Multivariate Probability Distributions Multivariate Probability Distributions Introduction to Statistics and Econometrics July 1, 2019 7/370 Lemma 1 (5.1). [Properties of 𝐹𝑋𝑌(𝑥, 𝑦)] Random Vectors and Joint Probability Distributions Random Vectors and Joint Probability Distributions
Multivariate Probability Distributions Random Vectors and Joint Probability Distributions Random Vectors and Joint Probability Distributions Theorem 1 (5.2).[Properties of Fxy(x,y)] Fx(x)=Fxy(x,+oo)and Fy(y)=Fxy(+o0,y). Proof: ·Define two events:A={X≤x},B={Y≤o}. Since B always holds,we have AnB=A.It follows that P(A)=P(AnB),that is,Fx(x)=Fxy(x,oo). Multivariate Probability Distributions Introduction to Statistics and Econometrics July1,2019 8/370
Multivariate Probability Distributions Multivariate Probability Distributions Introduction to Statistics and Econometrics July 1, 2019 8/370 Theorem 1 (5.2). [Properties of 𝐹𝑋𝑌(𝑥, 𝑦)] Random Vectors and Joint Probability Distributions Proof: Random Vectors and Joint Probability Distributions
Multivariate Probability Distributions Random Vectors and Joint Probability Distributions Random Vectors and Joint Probability Distributions Remarks: Theorem 5.2 implies that individual CDF's of X and Y can be obtained from the joint CDF Fxy(x,y). These individual CDF's are called the marginal CDF's of X and Y respectively. The joint and marginal CDF's are widely used in statis- tics and econometrics.An important example called cop- ula. Multivariate Probability Distributions Introduction to Statistics and Econometrics Juy1,2019 9/370
Multivariate Probability Distributions Multivariate Probability Distributions Introduction to Statistics and Econometrics July 1, 2019 9/370 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 Example 1 (5.1).[Bivariate Copula] Put U=Fx(X),V=Fy(Y).Then both the probability in- tegral transforms U and V are U[0,1]random variables.The joint CDF of (U,V) C(u,v)=P(U≤u,V≤v) is called the copula associated with the joint probability dis- tribution of (X,Y).The copula C(u,v)is closely related to the joint CDF Fxy(x,y). Multivariate Probability Distributions Introduction to Statistics and Econometrics July1,2019 10/370
Multivariate Probability Distributions Multivariate Probability Distributions Introduction to Statistics and Econometrics July 1, 2019 10/370 Example 1 (5.1). [Bivariate Copula] Random Vectors and Joint Probability Distributions Random Vectors and Joint Probability Distributions