If and are 2 random variables such that ~ (0,1) and ~ (0,1), and = min(,), prove that:
$1 - \mathbb{P}(X > \sqrt{z}) \mathbb{P}(Y > \sqrt{z}) - 1 + \mathbb{P}(X > -\sqrt{z}) \mathbb{P}(Y > - \sqrt{z})$ = $\mathbb{P}(X \leq \sqrt{z}) - \mathbb{P} (X \leq -\sqrt{z}) - \mathbb{P}(X \leq \sqrt{z}) \mathbb{P} (Y > \sqrt{z}) + \mathbb{P}(X \leq -\sqrt{z})\mathbb{P}(Y \leq \sqrt{z})$.
I'm having trouble proving the above expression. Any help is appreciated.