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Distribution of Log of Random Variable

  1. Apr 8, 2008 #1
    Let X and Y be random variables.

    X ~ N(u,s^2)
    Y = r ln X, where r is a constant.

    What is the distribution of Y?

    (This is not a homework problem. It's just related to something I was curious about, and I can't figure out how to solve this, if it is solvable...)
  2. jcsd
  3. Apr 8, 2008 #2


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    Homework Helper

    You know that

    [tex] 1 = \int_{-\infty}^{\infty}dx~\frac{1}{\sqrt{2\pi \sigma^2}} \exp\left[\left(\frac{x-\mu}{\sigma}\right)^2\right] = \int_{0}^{\infty}dx~\frac{2}{\sqrt{2\pi \sigma^2}} \exp\left[\left(\frac{x-\mu}{\sigma}\right)^2\right] [/tex]

    So, make a change of variables [itex]y = r \ln x[/itex]. The lower limit x = 0 becomes y = -\infty and the upper limit remains infinity. [itex]dy = r dx/x = r dx e^{-y/r}[/itex]


    [tex] 1 = \int_{-\infty}^{\infty}dy~\frac{2e^{y/r}}{r\sqrt{2\pi \sigma^2}} \exp\left[\left(\frac{e^{y/r}-\mu}{\sigma}\right)^2\right][/tex]

    The integrand is thus the probability density function for y. Note that the distribution is only valid for values of x zero or greater, as y is not defined for x < 0. This is why in the first line I used the evenness of the gaussian integrand to write it in terms of x > 0 only.
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