Multivariate Probability Distributions Random Vectors and Joint Probability Distributions The Discrete Case Remarks: .It is convenient to work on the support of (X,Y)only. Suppose x and y are the supports of X,Y respec- tively.Is it true that OxY 2x×y ={(x,y)∈R2:fx(x)>0,fy(y)>0}? Multivariate Probability Distributions Introduction to Statistics and Econometrics July1,2019 16/370
Multivariate Probability Distributions Multivariate Probability Distributions Introduction to Statistics and Econometrics July 1, 2019 16/370 Random Vectors and Joint Probability Distributions Remarks: The Discrete Case
Multivariate Probability Distributions Random Vectors and Joint Probability Distributions The Discrete Case ● Consider an example of bivariate distribution for which both X and Y takes nonnegative integers but with the restriction that X Y.Then we have 2x=2y={0,1,2,·} while 2xy={(x,y):0≤x≤y<o,x,y are integers} Obviously,nxy is a subset of xx ny. Multivariate Probability Distributions Introduction to Statistics and Econometrics Juy1,2019 171370
Multivariate Probability Distributions Multivariate Probability Distributions Introduction to Statistics and Econometrics July 1, 2019 17/370 Random Vectors and Joint Probability Distributions The Discrete Case
Multivariate Probability Distributions Random Vectors and Joint Probability Distributions The Discrete Case Question:Is it true that fxy(x,y)=1? (c,y)∈2xy ●For any subset A∈R2, P[(X,Y)∈A=fxy(,). (x,y)∈A Multivariate Probability Distributions Introduction to Statistics and Econometrics July1,2019 18/370
Multivariate Probability Distributions Multivariate Probability Distributions Introduction to Statistics and Econometrics July 1, 2019 18/370 Random Vectors and Joint Probability Distributions The Discrete Case
Multivariate Probability Distributions Random Vectors and Joint Probability Distributions The Discrete Case Example 2(5.2) Suppose X and Y have the joint PMF fxy(x,y)=cx+y for x =-1,0,1 and y=0,1, where c is an unknown constant. Find (1)the supports of X,Y,and (X,Y)respectively; (2)the value of c; (3)P(X=0andY=1); (4)P(X=1): (5)P(X-Y≤1). Multivariate Probability Distributions Introduction to Statistics and Econometrics Juy1,2019 19/370
Multivariate Probability Distributions Multivariate Probability Distributions Introduction to Statistics and Econometrics July 1, 2019 19/370 Random Vectors and Joint Probability Distributions The Discrete Case Example 2 (5.2)
Multivariate Probability Distributions Random Vectors and Joint Probability Distributions The Discrete Case Solution ● (1)We have2x={-1,0,1,2y={0,1},and2xy= {(-1,0),(0,1),(1,0),(1,1)}.Note that xy is a subset of2x×2 y because(0,0)∈2x×2ybut(0,0)2xy. ·(2)Using the property that∑xex∑y∈yfxY(r,)= 1,we have c[-1+0+|-1+1+0+0+0+1+1+0+1+1]=1. Thus,c=. To be Continued Multivariate Probability Distributions Introduction to Statistics and Econometrics Juy1,2019 20/370
Multivariate Probability Distributions Multivariate Probability Distributions Introduction to Statistics and Econometrics July 1, 2019 20/370 Random Vectors and Joint Probability Distributions The Discrete Case Solution To be Continued