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What is the pdf of multiplication of two random variables?

  1. Oct 28, 2012 #1
    We have two independent random variables X and Y whose pdfs are given as f(x) and f(y). Now when you multiply X and Y you get a random variable say Z. Now what is the resulting pdf f(z)?

    I mean how is that related to the pdf of f(x) and f(y)?

    From what I read it looks like

    f(z)=f(x) * f(y)

    where "*" represents convolution.

    But I couldn't find how you get that.
    Thanks a lot :)
  2. jcsd
  3. Oct 28, 2012 #2


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    Convolution is the operation for the sum of two variables.

    You can get fz via a double integral
    $$P(z<Z)=\iint_{x'y'=z} dx' dy' f_x(x') f_y(y')$$ where the integration limit is a hyperbola.
    You can split this into two parts with explicit integration limits like that:

    $$P(z<Z)=\int_{-\infty}^0 dx' f_x(x') \int_{z/x'}^\infty dy' f_y(y') + \int_0^\infty dx' f_x(x') \int_{-\infty}^{z/x'} dy' f_y(y')$$

    fz is the derivative of that.
  4. Oct 28, 2012 #3
    I know the meaning of convolution but what I would like to know is how multiplication of 2 random variables results in a pdf which is the convolution of the two pdfs.

    That's what I would like to know.

    Thanks a lot :)
  5. Oct 28, 2012 #4


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    It does not.
    But I gave one way to calculate the function in my post. You can even derive the whole equation (careful with the limits) to get a more direct expression for f(z).
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