Joint CDF (proof)

1. May 10, 2007

Zen

Last edited by a moderator: Apr 22, 2017 at 5:29 PM
2. May 10, 2007

happyg1

OK
So here's how we did this:
Let $$G_{x,y}=( \omega \in \Omega|X(\omega)\leq x,Y(\omega)\leq y)=(X\leq x,Y\leq y)$$
then
$$P(G_{x,y})=P(X\leq x,Y\leq y)=F(x,y)$$
then
$$P(a<X<b,c<Y<d)=G_{b,d}\setminus (G_{a,d}\cup G_{b,c})$$
then break up the RHS:
$$P(G_{b,d})-[P(G_{a,d})+P(G_{b,c})-P(G_{a,d}\cap G_{b,c})]$$
can you go from there?

Last edited: May 10, 2007
3. May 11, 2007

Zen

Thanks, I got it know. I also found the error in my original proof: P((B intersect D) U (A intersect C)) != 1