Normal and exponential-normal (?) distribution

by mahtabhossain
Tags: distribution, exponentialnormal, normal
 P: 2 Dear Users, For normally distributed random variables x and y's p.d.f.: $$\frac{1} {\sqrt{2\pi \sigma_x^2}}\exp\left\{- \frac{(x - \mu_x)^2}{2 \sigma_x^2}\right\}$$ and $$\frac{1} {\sqrt{2\pi \sigma_y^2}}\exp\left\{- \frac{(y - \mu_y)^2}{2 \sigma_y^2}\right\}$$ What will be the p.d.f. of ln(x) - ln(y)? Is there any method to find it? I think the followings are the p.d.f.s of u=ln(x) and v=ln(y) given x and y are normally distributed: $$\frac{1} {\sqrt{2\pi \sigma_x^2}}\exp\left\{u - \frac{(e^u - \mu_x)^2}{2 \sigma_x^2}\right\}$$ and $$\frac{1} {\sqrt{2\pi \sigma_y^2}}\exp\left\{v - \frac{(e^v - \mu_y)^2}{2 \sigma_y^2}\right\}$$ I am trying to find the p.d.f. of ln(x)-ln(y). Any suggestions?