# Thread: joint pdf and independence

1. ## joint pdf and independence

X~Gamma(theta,alpha)
y~Gamma(theta,beta)
U=X/Y
V=X+Y

Find fX,Y(x,y)
State whether U and V are independent
Find marginal pdf

i am having trouble finding fX,Y(x,y) using information given that X and Y are of Gamma distribution.

And what is the theorem used to determine the independence of U and V?

2. Originally Posted by appleting
X~Gamma(theta,alpha)
y~Gamma(theta,beta)
U=X/Y
V=X+Y

Find fX,Y(x,y)
State whether U and V are independent
Find marginal pdf

i am having trouble finding fX,Y(x,y) using information given that X and Y are of Gamma distribution.

And what is the theorem used to determine the independence of U and V?
Are X and Y independent?

If U and V are independent then the joint pdf g(u, v) can be written as $g(u, v) = g_U(u) \cdot g_V(v)$.

Use the change-of-variables formula to find g(u, v). Obviously f(x, y) is needed first, whch means that the answer to my quesiton is needed.

3. Originally Posted by mr fantastic
Are X and Y independent?

If U and V are independent then the joint pdf g(u, v) can be written as $g(u, v) = g_U(u) \cdot g_V(v)$.

Use the change-of-variables formula to find g(u, v). Obviously f(x, y) is needed first, whch means that the answer to my quesiton is needed.
oops! yes, X and Y are independent!

4. Originally Posted by appleting
oops! yes, X and Y are independent!
OK. So the joint pdf of X and X should be obvious. Then you can use the change-of-variables formula to get the joint pdf of U and V, g(u, v).

Then see whether of not g(u, v) can be written as a product $g(u, v) = g_U(u) \cdot g_V(v)$. If yes, then U and V are independent. If no, then they're not.

5. Originally Posted by mr fantastic
OK. So the joint pdf of X and X should be obvious. Then you can use the change-of-variables formula to get the joint pdf of U and V, g(u, v).

Then see whether of not g(u, v) can be written as a product $g(u, v) = g_U(u) \cdot g_V(v)$. If yes, then U and V are independent. If no, then they're not.
hmm im having problem with the gamma distribution in this question

fx,y(X,Y) = {[(theta)^(alpha+beta)][x^(alpha-1)][y^(beta-1)][e^(-theta*(x+y))]}/[T(alpha+beta)] ??

or is it

fx,y(X,Y) = {[(theta)^(alpha+beta)][x^(alpha-1)][y^(beta-1)][e^(-theta*(x+y))]}/[T(alpha)T(beta)] ??

or am i completely wrong? help~

(T is that sign thats like a an upside down mirrored L)

6. Originally Posted by appleting
hmm im having problem with the gamma distribution in this question

fx,y(X,Y) = {[(theta)^(alpha+beta)][x^(alpha-1)][y^(beta-1)][e^(-theta*(x+y))]}/[T(alpha+beta)] ??

or is it

fx,y(X,Y) = {[(theta)^(alpha+beta)][x^(alpha-1)][y^(beta-1)][e^(-theta*(x+y))]}/[T(alpha)T(beta)] ??

or am i completely wrong? help~

(T is that sign thats like a an upside down mirrored L)
What you need is here: Gamma distribution - Wikipedia, the free encyclopedia. Multiply the pdf's of X and Y together.