# Joint density proofing

• Oct 16th 2012, 04:03 PM
cummings123321
Joint density proofing
if (X,Y) is a random vector in R^2,then define f(X,Y) to be its joint density if
P((X,Y)∈A)=∫f(X,Y)(x,y)dxdy, for all reasonable sets A.

show if (X,Y) is
a random vector with density f(X,Y) and f(X,Y)(x,y)=f(x)g(x) for a pair
non-negative functions f and g then X has density f/(∫f(t)dt) and Y has density
g/(∫g(t)dt) and X and Y are independent.(hints: shat means showing P((X∈A)∩
(X∈B))=P(X∈A)P(X∈B)

can someone help me to construct the proof?
• Oct 16th 2012, 05:47 PM
chiro
Re: Joint density proofing
Hey cummings123321.

What is the definition of the marginal density functions for X and Y in terms of the joint distribution? You need to use the relationship between these and the joint to show the independence criterion (i.e P(A and B) = P(A)P(B)).
• Oct 17th 2012, 01:14 AM
cummings123321
Re: Joint density proofing
I don't quite undetstand why the density of X is f/(∫f(t)dt, now i know is MDF are f(x)=∫f(x)g(x)dy and f(x)=∫f(x)g(x)dx , how should i get the density from MDF to PDF??
• Oct 17th 2012, 01:34 AM
chiro
Re: Joint density proofing
Can you link the product of the X and Y marginals to the joint distribution?