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Transformations and joint pdf's

  1. Jun 6, 2012 #1
    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,

  2. jcsd
  3. Jun 6, 2012 #2


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    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.
  4. Jun 6, 2012 #3
    I think I see what you mean
    so fy(y)= f( g-1(y1,y2)) . Jacobian[ g-1(y1,y2)]
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