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Distribution of Product of Dependent RV's

  1. Aug 18, 2010 #1
    Distribution of Product of Dependent RV's

    Hello all.
    Let's say we have two random variables, say X and Y.
    We know the marginal densities for them, say Px(X) and Py(Y).
    How do we find the density of Z = X*Y?
    The important part here is that X and Y are dependent.
    If there are any tips or directions you can point me then great.
     
  2. jcsd
  3. Aug 19, 2010 #2
    An application of the product formula is given here.

    http://www.math.wm.edu/~leemis/2003csada.pdf

    If X and Y are dependent then f X,Y (x,y)=f Y|X (y|x) f X(x)=f X|Y (x|y) f Y(y)

    Note: E[X,Y]=E[X]E[Y]+Cov [X,Y]
     
    Last edited: Aug 19, 2010
  4. Aug 19, 2010 #3
    Thanks for the reply. However, note that this paper shows an algorithmic approach to the calculation of the distribution of independent random variables.

    From the abstract,
    What I need is an understanding of the case when the variables in question are dependent.
     
  5. Aug 19, 2010 #4
    Also I am not saying that I need to find the joint density of X and Y.

    I need to think about the distribution of Z, when Z = X*Y and all I have to go on are X and Y's marginals.
     
  6. Aug 19, 2010 #5
    The Rohatgi integral can handle dependence. That's why I included the formula for f X,Y (x,y) when X and Y are dependent.

    Also see:http://en.wikipedia.org/wiki/Product_distribution
     
    Last edited: Aug 19, 2010
  7. Aug 19, 2010 #6
    I see. So we will be needing the joint distribution.
    My mistake, thank you very much for your help.
     
  8. Aug 19, 2010 #7
    You're welcome.
     
  9. Aug 20, 2010 #8
    Error in post 2: That should be E[XY]=E[X]E[Y]+Cov[X,Y]
     
    Last edited: Aug 20, 2010
  10. Oct 15, 2010 #9
    Hello again
    Can you give tips on also distribution of:
    sum or difference on random variables that are
    -possibly dependent
    -non Gaussian
    Thank you
     
  11. Oct 16, 2010 #10
    I got it, it is
    Z=X+Y
    [tex] f_Z(z)=\int_{-\infty}^{\infty}f(x,z-x)dx [\tex]

    where f is the joint dist
     
  12. Oct 16, 2010 #11
    corrected Latex
     
  13. Oct 17, 2010 #12
    Arrg. Thanks Matey

    [tex] f_Z(z)=\int_{-\infty}^{\infty}f(x,z-x)dx [/tex]

    Shiver me timbers
    How does one edit an old post? thanks
     
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