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.
"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)"
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