Characterizing the distribution $\max\{X,Y\}/X$ for uniform $X,Y$

Question

Let X and Y independent random variables with uniform distribution on (0,1).

Let $Z=\frac{\max\left \{ X,Y \right \}}{X}$

a) Calculate $P(Z=1)$ , and observe that $Z$ is not a continuous variable

b) Find the distribution function of $Z$ and observe that $Z$ is not a discrete variable

Answer 1

HINT

$$ Z = 1 \iff X = \max\{X,Y\} \iff X \ge Y. $$

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