Multivariate Probability Distributions Random Vectors and Joint Probability Distributions The Discrete Case (3)By the definition of the joint PMF,we have PX=0,y=)=1y0,)=0+1=号 (4)Noting that the event =1==1nreny,we have P(X 1) ∑fxY(1,) y∈2Y fxy(1,0)+fxy(1,1) 3 5 To be Continued Multivariate Probability Distributions Introduction to Statistics and Econometrics Juy1,2019 21/370
Multivariate Probability Distributions Multivariate Probability Distributions Introduction to Statistics and Econometrics July 1, 2019 21/370 Random Vectors and Joint Probability Distributions The Discrete Case To be Continued
Multivariate Probability Distributions Random Vectors and Joint Probability Distributions The Discrete Case ·(5) P(IX-Y\≤1)= ∑ fxy(x,y) (x,y)∈2xy:lx-yl≤1 =fxy(-1,0)+fxy(0,1)+fxy(1,0)+fxy(1,1) 1. Multivariate Probability Distributions Introduction to Statistics and Econometrics Juy1,2019 22/370
Multivariate Probability Distributions Multivariate Probability Distributions Introduction to Statistics and Econometrics July 1, 2019 22/370 Random Vectors and Joint Probability Distributions The Discrete Case
Multivariate Probability Distributions Random Vectors and Joint Probability Distributions The Discrete Case Remarks: What is the relationship between fxy(x,y)and Fxy(x,y)? For DRV's X and Y,we can obtain the joint CDF Fxy(x,y)=P(X≤x,Y≤y) ∑fxy(u,), (u,v)∈2xy(c,y) where xy(,y)is the set of all possible pairs of (u,v)in the support xy of (X,Y)such that u x,v <y,namely, 2xY(x,y)={(u,v)∈2xy,w≤r,v≤y}. Thus,we can obtain Fxy(a,y)from fxy(x,y). Multivariate Probability Distributions Introduction to Statistics and Econometrics July1,2019 23/370
Multivariate Probability Distributions Multivariate Probability Distributions Introduction to Statistics and Econometrics July 1, 2019 23/370 Random Vectors and Joint Probability Distributions Remarks: The Discrete Case
Multivariate Probability Distributions Random Vectors and Joint Probability Distributions The Discrete Case By taking the differences of Fxy(,y)with respect to x and y,we can also recover fxy(x,y)from Fxy(x,y). Without loss of generality,we assume that the possible values X:1<x2 x3< ·,and the possible values Y:y1<y2<y3<··.Then,fori>1,j>1, fxy(xi,Vj) 三 △y△xFxY(x,y） 三 △y[FxY(x,y）-FxY(x-1,y)] [Fxy(xi,yj)-Fxy(i,Uj-1)]-[Fxy(xi-1,Uj)-Fxy(xi-1,Uj-1) Fxy(xi,yj)-Fxy(i,Uj-1)-Fxy(i-1,4j)+FxY(i-1,9j-1), where Ax and Ay are the difference operators with respect to x and y respectively. Multivariate Probability Distributions Introduction to Statistics and Econometrics July1,2019 24/370
Multivariate Probability Distributions Multivariate Probability Distributions Introduction to Statistics and Econometrics July 1, 2019 24/370 Random Vectors and Joint Probability Distributions The Discrete Case
Multivariate Probability Distributions Random Vectors and Joint Probability Distributions The Discrete Case .The above formula does not cover the cases where i=1 or j=1.For these cases,we have Fxy(i,4j)-Fxy(xi,yj-1),if i=1,j>1, (r)- Fxy(xi,Uj)-Fxy(ai-1,Uj), fi>1,j=1, Fxy(xi,Vj), fi=1,j=1. The bivariate concepts can be generalized to the multivariate case.For example,the joint PMF of n discrete random variables X1,...,Xn is given by fxn(x")=P(X1=21,..,Xn =En) for each n tuple x"=(1,..,n)E R",and the joint CDF of Xm= (X1,...,Xn)is given by Fxn(x")=P(X1≤x1,·,Xn≤xn) for all xT∈Rn Multivariate Probability Distributions Introduction to Statistics and Econometrics July1,2019 25/370
Multivariate Probability Distributions Multivariate Probability Distributions Introduction to Statistics and Econometrics July 1, 2019 25/370 Random Vectors and Joint Probability Distributions The Discrete Case