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Math Help - A problem from "First course in Probability"-Ross

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    A problem from "First course in Probability"-Ross

    Given that X and Y are independent continuous positive random variables, express the density of the random variable Z in terms of the density of X and Y in each case.
    a) Z = X/Y

    b) Z = XY

    c) Given that X~exp(lamda), Y~exp(u), evaluate the density obtained in part a)
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  2. #2
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    Quote Originally Posted by dingdong View Post
    Given that X and Y are independent continuous positive random variables, express the density of the random variable Z in terms of the density of X and Y in each case.
    a) Z = X/Y

    b) Z = XY

    c) Given that X~exp(lamda), Y~exp(u), evaluate the density obtained in part a)
    c) Something to get you started (and which should also give you some food for thought for a) and b)):

    The cdf of Z is given by

    F(z) = \Pr(Z < z) = \Pr\left( \frac{X}{Y} < z \right) = \Pr(X < zY)

    = \int_{y=0}^{y = + \infty} \int_{x=0}^{x = zy} \lambda \, \mu \, e^{-\lambda x} \, e^{-\mu y} \, dx \, dy, 0 \leq z < +\infty.

    The integrals should be routine at this level.

    The pdf of Z is given by f(z) = \frac{dF}{dz}.
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