Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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


    User Avatar
    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.
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?

Similar Discussions: Distribution of Log of Random Variable
  1. Random variable x (Replies: 3)