$\max\{U_1(0,1),U_2(0,1)\}$ fits what distribution?

Question

$U_1$ and $U_2$ are i.i.d. random variables from U(0,1). I want to know why $\max\{U_1,U_2\} \sim RT(0,1)$ The cdf of RT(0,1) is as follows.

Answer 1

Lets define $F = \max\{U_1, U_2\}$ and find the probability $P(F < x)$. Obviously, $P(F < x) = P(\max\{U_1, U_2\} < x) = P(U_1 < x, U_2 < x) = P(U_1 < x)P(U_2 < x).$

For $x < 0$ we get 0, for $x > 1$ we get 1. And for $0 \le x \le 1$ we see $P(U_1 < x) = x, P(U_2 < x) = x$. So, $P(F < x) = x^2$.