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Math Help - Gamma variables and transformations

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    Gamma variables and transformations

    If Y is uniformly distributed over the interval (0,1), i.e. f(y) = 1 for 0< y < 1.
    Show that U = -2 loge(Y) has a negative exponential distribution. By comparing the density
    functions or the moment generating functions, show also that an exponential distribution is
    identical to a chi-square distribution with 2 degrees of freedom. If Y1, Y2, ..., Yn are
    independently uniformly distributed over the interval (0,1), determine the distribution of
    U = -2loge(Y1Y2...Yn) = summation(-2loge(Yi)).
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    Quote Originally Posted by sonitgarg View Post
    If Y is uniformly distributed over the interval (0,1), i.e. f(y) = 1 for 0< y < 1.
    Show that U = -2 loge(Y) has a negative exponential distribution. By comparing the density
    functions or the moment generating functions, show also that an exponential distribution is
    identical to a chi-square distribution with 2 degrees of freedom. If Y1, Y2, ..., Yn are
    independently uniformly distributed over the interval (0,1), determine the distribution of
    U = -2loge(Y1Y2...Yn) = summation(-2loge(Yi)).
    Here's a start:

    Calculate the cdf of U:

    F(u) = \Pr(U < u) = \Pr(-2 \ln Y < u) = \Pr\left(\ln Y > -\frac{u}{2}\right)  = \Pr(Y > e^{-u/2}) = \int^{1}_{e^{-u/2}} dy = .... if u > 0 and zero otherwise.

    Now recall that the pdf of U is given by f(u) = \frac{dF}{du}.
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