Probability density function of two uniformly distributed stochastic variables

Question

I'm currently stuck on an exercise involving two independent stochastic variables X and Y.

Both X and Y ~ U(0,1) (uniform distribution)

The goal of the exercise is to calculate the probability density function of $$ Z =\ln{ \bigg(\cfrac{\max(X,Y)}{\min(X,Y)}}\bigg) $$ I got stuck and looked at the solutions manual, and don't seem to understand how $$ P \bigg(\ln{ \bigg(\cfrac{\max(X,Y)}{\min(X,Y)}}\bigg) \leq z \bigg) $$

can be rewritten as

$$ P({Y \leq e^z \cdot X} \cap { X \leq Y}) + P({X \leq e^z \cdot Y} \cap { Y\leq X}) $$

Thanks in advance!

Answer 1

You get $$ \log\frac{X\vee Y}{X\wedge Y} \leq z \iff X\vee Y \leq \mathrm e^z(X\wedge Y) $$ $$ \iff X\leq\mathrm e^z Y,Y\leq X \text{ or }Y\leq\mathrm e^z X,Y\geq X $$ where $\wedge$ if for $\min$, $\vee$ is for $\max$, and the latter two events have intersection of zero probability.