Deriving the logistic distribution from two exponential distributions. Is there no division by zero?

Question

How do I prove that if $X, Y \sim \operatorname{Exponential}(1)$ then $\mu-\beta\log\left(\frac X Y \right) \sim \operatorname{Logistic}(\mu,\beta)$?

Shouldn't it always cause a division by zero?

I feel totally naïve here!

Answer 1

Let $Z = \frac{X}{Y}$.

Then $F_Z(z) = \Pr[Z \leq z] = \Pr[X/Y \leq z] = \int_0^\infty \Pr[X \leq yz]f_{Y}(y) dy = \int_0^\infty F_X(yz) f_{Y}(y) dy$

You can now calculate $f_Z$ from here. Does that help?