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Density function of product of random variables

  1. May 7, 2014 #1
    suppose you have two random variables X and Y which are independent,
    we want to form a new random variable Z=XY, if f(x) and f(y) are density functions
    of X and Y respectively what is the density function of Z?

    I tried taking logs and applying convolution, but it did not really work
  2. jcsd
  3. May 7, 2014 #2


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    Homework Helper

    It isn't clear to me from your writing whether X and Y have the same distribution - you used the same name for their densities - but think about this start: it will work whether they are identically distributed or not. Start with this
    P(Z \le z) = E_Y \left[P\left( XY \le z \mid Y = y\right) \right]

    and see what develops.
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