Homework Help: Transformations and joint pdf's

1. Jun 6, 2012

6.28318531

1. The problem statement, all variables and given/known data

Let X1 and X2 be random variables having a joint pdf, fX1X2(x1,x2). Suppose that Y1=X1X2, and Y2=X1X2 Use the transformation result to derive an expression for the joint pdf of Y1 and Y2
in terms of that for X1 and X2
2. Relevant equations

The single random variable case

fy(y)=f[g-1(y)] |dg-1(y)/dy|
where g is our transformation

3. The attempt at a solution
So many subscripts,

Anyway I know the single variable case, so how do I generalise this to multiple random variables? Do much the same thing? Let g(Y1,Y2)= (X1 X2,X1/X2) , then what take ∇ .g-1? I'm not really sure how you generalise the derivative part,

Thanks,

2. Jun 6, 2012

vela

Staff Emeritus
Think back to calculus when you changed variables from x and y to u=u(x,y) and v=v(x,y) in 2-dimensional integrals. You're doing the same thing here. You need to use the Jacobian.

3. Jun 6, 2012

6.28318531

I think I see what you mean
so fy(y)= f( g-1(y1,y2)) . Jacobian[ g-1(y1,y2)]