Question

A random vector (X1, X2) is a discrete random vector if its space D or range space is finite or countable. Suppose X1, X2 are positive integer valued and we need to find P(X1= x1, X2= x2), where x1, x2 are known positive integers, in terms of their known cumulative distribution function F.

Answer 1

$X_1, X_2$ are discrete random variables

$F_{X_1,X_2}(x_1,x_2) = P(X_1 ≤x_1, X_2≤x_2) \\ P(X_1 = x_1, X_2=x_2) = P(X_1 ≤x_1, X_2≤x_2) -P(X_1 <x_1, X_2<x_2) \\ = P(X_1 ≤x_1, X_2≤x_2) -P(X_1 ≤x_1-1, X_2≤x_2-1) \\ = F(x_1,x_2) -F(x_1- 1, x_2 -1)$