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\label{PROB-2}Let [math]X[/math], [math]Y\sim\mathcal{N}(0,1,\mathbb{R}^d)[/math]. Show the following.

• [math]\forall\:d\geqslant1\colon\E(\|X-Y\|-\sqrt{2d})\leqslant1/\sqrt{2d}[/math].
• [math]\forall\:d\geqslant1\colon\V(\|X-Y\|)\leqslant 3[/math].

\smallskip { \small Hint: Check firstly [math]\V((X_i-Y_i)^2)=3[/math] by establishing that [math]X_i-Y_i\sim\mathcal{N}(0,2,\mathbb{R})[/math] and by using a suitable formula for computing the fourth moment. Conclude then that [math]\V(\|X-Y\|^2)\leqslant3d[/math]. Adapt finally the arguments we gave above for [math]\E(\|X\|-\sqrt{d})[/math] and [math]\V(\|X\|)[/math]. }