What is the pdf of $Z=X/\max(X,Y)$ with $X,Y$ exponentials of lambda parameter?

Question

Given $X,Y$ 2 independent r.v.'s both distributed as $\exp(λ)$, what is the pdf of $Z=X/\max(X,Y)$?

Answer 1

For $0 < t < 1$, \begin{align} P(Z < t) =& P(X < t \max(X,Y)) \\ =& P(Y>X \text{ and }X < tY) \\ =& P(X < tY) \\ =& \int_{0}^{+\infty} \lambda e^{-\lambda y}(\int_{0}^{ty }\lambda e^{-\lambda x} dx)dy \\ =& \frac{t}{1+t} \end{align}

and $P(Z = 1) = P(X \geq Y) = \frac{1}{2}$.

Here we get the cdf, since it's not absolutely continuous with respect to the Borel measure, there is no pdf.