Find E(max({X^2,Y^2})) of two discrete random variables

Question

I don't understand how to arrive at this result, is my first attempt at this kind of exercise and i'm really confused, iìm trying to using basic formulas as: Cov(X,Y) = E(X,Y) - E(X)E(Y), Var(X) = E(X^2) - E(X)^2 and so on, but i stil can't arrive at any result

Answer 1

For all $x,y\in\mathbb R$, we have $\max(x,y)=\frac{x+y}{2}+\frac{\vert x-y\vert}{2}$. Therefore, using Cauchy-Scwharz inequality, we have \begin{align*} \mathbb E[\max(X^2,Y^2)]&=1+\frac12\mathbb E[\vert X^2-Y^2\vert]\\ \mathbb E[\vert X^2-Y^2\vert]&=\mathbb E[\vert X-Y\vert]\mathbb E[\vert X+Y\vert]\le\sqrt{\mathbb E[\vert X-Y\vert^2]\mathbb E[\vert X+Y\vert^2]}\\ &=\sqrt{(2-2\rho)(2+2\rho)}=\sqrt{4-4\rho^2}=2\sqrt{1-\rho^2} \end{align*}