厦门大学：《概率论与数理统计 Probability and Statistics for Economists》课程教学资源（课件讲稿）Chapter 05 Multivariate Probability Distributions

养院究等濟装南型王季大門厦餐 Multivariate Probability Distributions Professor Yongmiao Hong Cornell University Juy1,2019

Multivariate Probability Distributions Professor Yongmiao Hong Cornell University July 1, 2019

CONTENTS 5.1 Random Vectors and Joint Probability Distributions 5.2 Marginal Distributions 5.3 Conditional Distributions 5.4 Independence 5.5 Bivariate Transformation 5.6 Bivariate Normal Distribution 5.7 Expectations and Covariance 5.8 Joint Moment Generating Function 5.9 Implications of Independence on Expectations 5.10 Conditional Expectations 5.11 Conclusion Multivariate Probability Distributions Introduction to Statistics and Econometrics Juy1,2019 2/370

Multivariate Probability Distributions Introduction to Statistics and Econometrics July 1, 2019 2/370 5.1 Random Vectors and Joint Probability Distributions 5.2 Marginal Distributions 5.3 Conditional Distributions 5.4 Independence 5.5 Bivariate Transformation 5.6 Bivariate Normal Distribution 5.7 Expectations and Covariance 5.8 Joint Moment Generating Function 5.9 Implications of Independence on Expectations 5.10 Conditional Expectations 5.11 Conclusion CONTENTS

Multivariate Probability Distributions Random Vectors and Joint Probability Distributions Random Vectors and Joint Probability Distributions Definition 1(5.1).[Random Vector] An n-dimensional random vector,denoted as z=(Z1,.. Zn)',is a function from a sample space S into R",the n- dimensional Euclidean space.For each outcome se S,Z(s)is an n-dimensional real-valued vector and is called a realization of the random vector Z. Multivariate Probability Distributions Introduction to Statistics and Econometrics July1,2019 3/370

Multivariate Probability Distributions Multivariate Probability Distributions Introduction to Statistics and Econometrics July 1, 2019 3/370 Definition 1 (5.1). [Random Vector] 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 Remarks: In this chapter,we will mainly focus on bivariate prob- ability distributions,which can illustrate most (but not all)essentials of multivariate probability distributions. We shall consider two random variables (X,Y)in most of the subsequent discussion,where both X and Y are defined on the same probability space (S,B,P). A realization of (X,Y)will be a pair (x,y)E R2. Multivariate Probability Distributions Introduction to Statistics and Econometrics July1,2019 41370

Multivariate Probability Distributions Multivariate Probability Distributions Introduction to Statistics and Econometrics July 1, 2019 4/370 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 Definition 2 (5.2).[Joint CDF] The joint CDF of X and Y is defined as follows: Fxy(x,y) P(X≤x,Y≤y) P(X≤x∩Y≤y) for any pair(x,y)∈R2. Multivariate Probability Distributions Introduction to Statistics and Econometrics Juy1,2019 5/370

Multivariate Probability Distributions Multivariate Probability Distributions Introduction to Statistics and Econometrics July 1, 2019 5/370 Definition 2 (5.2). [Joint CDF] Random Vectors and Joint Probability Distributions Random Vectors and Joint Probability Distributions

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