I have three random variables, $X, Y, Z$. I know that
$$\log X - \log Y - \log Z = \log \frac{X}{YZ} \sim \mathcal{N}(0, \sigma^2)$$
How can I recover the distribution of $\frac{X}{YZ}$? Is it as simple as $e^{\mathcal{N}(0, \sigma^2)}$?
$\begingroup$
$\endgroup$
asked May 2, 2014 at 4:15
Add a comment
|
1 Answer
$\begingroup$
$\endgroup$
4
Let $W = \log X - \log Y - \log Z$.
So $W\sim N(0,\sigma^2)\,$.
What's the distribution of $\exp(W)$?
If you don't know what this is already, you can recover the density by simple change of variable.
If $Y=\phi(X)$, then:
$p_y(y) = p_x(\phi^{-1}(y)) ~ \left|\frac{d\phi^{-1}}{dy}\right|. $
Edit:
As the OP has now discerned, this means that $\exp(W)=\frac{X}{YZ}$ is $\text{lognormal}(0,\sigma^2)$.
-
$\begingroup$ So in this case, $\phi$ is $\exp$ and $E$ is $W$ and so $p_y(y) = p_W(\exp^{-1}(y)) |\frac{d \exp^{-1}}{dy}|$ ... ? $\endgroup$ Commented May 2, 2014 at 5:31
-
$\begingroup$ Well, $X$ is $W$ and $Y$ is $\exp(W)$. Further, you know what the inverse function of $\exp$ is already. $\endgroup$– Glen_bCommented May 2, 2014 at 5:49
-
$\begingroup$ Oh. So exp(W) is log-normally distributed then? $\endgroup$ Commented May 2, 2014 at 5:59
-