Distribution of Log of Random Variable

[andrewcheong]

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...)

 

[Mute]

You know that

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]

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

Hence,

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]

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.