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Math Help - Product of two random variables

  1. #1
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    Product of two random variables

    Hi guys,

    Need a hint with the following problem:

    Let X and Y be independent random variables, with probability distribution functions f_{X}(x) and f_{Y}(y). Let Z= XY What is the probability distribution function for Z?

    F_{Z}(x) =P(Z \leq z) = P(XY \leq z) = P (Y \leq \frac{z}{x}) = \int_{-\infty}^{\infty}\int_{-\infty}^{\frac{z}{x}}f_{XY}(x, y)\,dy\,dx
    Since X and Y are independent, this gives:

    F_{Z}(z) = \int_{-\infty}^{\infty}\int_{-\infty}^{\frac{z}{x}}f_{X}(x)f_{Y}(y)\,dy\,dx

    Thus, we can obtain f_{Z}(z) by differentiating both sides w/ respect to z and using the fundamental theorem of calculus, which gives

    f_{Z}(z) = \int_{-\infty}^{\infty}\frac{1}{x}f_{X}(x) g_{Y}(\frac{x}{z})\,dx

    Thanks any help!

    James
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  2. #2
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    Re: Product of two random variables

    Quote Originally Posted by james121515 View Post
    Hi guys,

    Need a hint with the following problem:

    Let X and Y be independent random variables, with probability distribution functions f_{X}(x) and f_{Y}(y). Let Z= XY What is the probability distribution function for Z?

    F_{Z}(x) =P(Z \leq z) = P(XY \leq z) = P (Y \leq \frac{z}{x}) = \int_{-\infty}^{\infty}\int_{-\infty}^{\frac{z}{x}}f_{XY}(x, y)\,dy\,dx
    Since X and Y are independent, this gives:

    F_{Z}(z) = \int_{-\infty}^{\infty}\int_{-\infty}^{\frac{z}{x}}f_{X}(x)f_{Y}(y)\,dy\,dx

    Thus, we can obtain f_{Z}(z) by differentiating both sides w/ respect to z and using the fundamental theorem of calculus, which gives

    f_{Z}(z) = \int_{-\infty}^{\infty}\frac{1}{x}f_{X}(x) g_{Y}(\frac{x}{z})\,dx

    Thanks any help!

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

    Product distribution - Wikipedia, the free encyclopedia
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  3. #3
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    Re: Product of two random variables

    Hey,

    Interesting, you have the same question as I have!
    http://www.mathhelpforum.com/math-he...es-192173.html

    I'm specifically looking for the case where X is exponentially distributed, and Y follows a normal distribution.

    A closed-form solution seems impossible, hence I tried to estimate the pdf of Z.

    Via Monte Carlo simulation, it appears that Z=XY, follows again an exponential distribution for z>0.
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