Transformations and joint pdf's

6.28318531

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

Let X₁ and X₂ be random variables having a joint pdf, f[SUB]X¹X₂[SUB](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

f_y(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,

vela

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.

6.28318531

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