How to prove the cumulative distribution function of the maximum of a pair of values drawn from random variable X is the square of the original CDF Suppose $X=Y^2$, where $Y\sim U(0,1)$. Now we need to find the cumulative distribution function of $X$.
$$P(X<x)=P(Y^2<x)=P(Y<x\sqrt x)=F_Y(x\sqrt x)$$
That's all?

If $X$ and $Y$ are independent and identically distributed then
$P(\max(X,Y) \le z)$
$= P(X \le z \text{ and } Y \le z) $
$= P(X\le z) P(Y \le z) \text{ because they are independent}$
$= \left(P(X\le z)\right)^2 \text{ because they are identically distributed}$
