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