Distribution of the minimum of two uniform random variates

Question

Suppose $X,Y\sim U(0,1)$ are iid random variables and $Z=\min(X,Y)$. Find the pdf and expected value of $Z$.

I've worked this out before when $Z=\max(X,Y)$, but I can't even start here with the maximum replaced with the minimum. Any help?

Answer 1

For $\max\{X,Y\}$ the key is that $\mathbb{P}(\max\{X,Y\}\leq z)=\mathbb{P}(X\leq z,Y\leq z)$.

So for the minimum, what can you say about $\mathbb{P}(\min\{X,Y\}>z)$?