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Explain p.d.f. of the sum of random variables

  1. Oct 13, 2012 #1
    I need your help,
    Say we have two random variables with some joint pdf f(x,y). How would I go about finding the pdf of their sum?
  2. jcsd
  3. Oct 13, 2012 #2


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    If the two random variables are dependent there is no easy solution. (I tried Google!)
  4. Oct 13, 2012 #3
    I think I found the answer: google docs... page 10

    another question, if I end up with an integral like that and a constant inside, then does the pdf of the two variables diverges and there's no answer?
  5. Oct 14, 2012 #4


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    Hey exoCHA.

    The PDF won't diverge if it's a valid PDF.

    The PDF however may not give finite moments but that is a completely different story. Your PDF should be a valid PDF if you did everything right and it should contain a region that has finite realizations of Z if the domain of your X and Y in your joint PDF are also finite for something like Z = X + Y.

    Remember that even a Normal distribution is valid for the entire real line just like the Z will also be valid across the entire real line.
  6. Oct 14, 2012 #5


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    Another way to find the pdf would be to calculate the joint characteristic function

    $$\Gamma(\mu_x,\mu_y) = \langle e^{i\mu_x x + i\mu_y y} \rangle = \int_{-\infty}^\infty dx \int_{-\infty}^\infty dy~\rho(x,y) e^{i\mu_x x + i\mu_y y}.$$

    One can recover the joint pdf by inverse transforming in the two different mu's, but you can also find the pdf of their sum by setting both ##\mu_x=\mu_y = \mu## and inverse fourier transforming in ##\mu##:

    $$\rho_Z(z) = \int_{-\infty}^\infty \frac{d\mu}{2\pi} \Gamma(\mu,\mu)e^{-i\mu z},$$
    where z = x + y. (Whether or not this integral is easy to do is of course a separate issue).
  7. Oct 14, 2012 #6


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    If you have a density function f(x,y), then the probability that X+Y < z can be expressed as a double integral,
    ∫∫f(x,y)dxdy where the integral range is given by x+y < z. This can be evaluated by u=x+y replacing x, so du = dx. The double integral now has y range (-∞,∞) and u range (-∞,z) and g(u,y) = f(u-y,y) as the integrand.
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