Independence of Bivariate Continuos RV's.

[Actuarial_Deepika]

By the two definitions of independence for bivariate continuous RVs: (1) F(x,y)=F_X(x)F_Y(y) and
(2) f(x,y)=f_X(x)f_Y(y).
Prove that these two are equivalent. That is: prove that (1) implies (2) and that (2) implies (1).

I tried to differentiate for one and integrate for the other.

 

[BGM]

Yes your approach is correct, so what is your problem?

 

[Actuarial_Deepika]

I am unable to get the specified proof.

 

[Dason]

I am unable to get the specified proof.

Yeah. We inferred that. The problem we're having is that you're literally giving us no information about what is giving you trouble.

 

[Actuarial_Deepika]

When I integrate it, what limits should i take?

 

[BGM]

Actually just a few steps: following from the definition

\( F_{X,Y}(x, y) = \Pr\{X \leq x, Y \leq y\} \)

and almost finish the question.

 

[Actuarial_Deepika]

Thank you so much! You are a life saver!

 

[BGM]

You are welcome. But I just wonder you do not have the definition in your textbook?

 

[Actuarial_Deepika]

Actually, I don't have the text book. I am trying to gather all the information off the internet.