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

$P(X_1= x_1, X_2= x_2)=P(X_1= x_1, X_2\leq x_2)-P(X_1= x_1, X_2< x_2)$

$=P(X_1= x_1, X_2\leq x_2)-P(X_1= x_1, X_2\leq x_2-1)$

$P(X_1= x_1, X_2\leq x_2)=P(X_1\leq x_1, X_2\leq x_2)-P(X_1< x_1, X_2\leq x_2)$

$=P(X_1\leq x_1, X_2\leq x_2)-P(X_1\leq x_1-1, X_2\leq x_2)$

$=F(x_1,x_2)-F(x_1-1,x_2)$

Similarly, $P(X_1= x_1, X_2\leq x_2-1)=F(x_1,x_2-1)-F(x_1-1,x_2-1)$.

Therefore,

$P(X_1= x_1, X_2= x_2)=$

$=(F(x_1,x_2)-F(x_1-1,x_2))-(F(x_1,x_2-1)-F(x_1-1,x_2-1))$

$=F(x_1,x_2)-F(x_1-1,x_2)-F(x_1,x_2-1)+F(x_1-1,x_2-1)$