exercise:093e852bc9: Difference between revisions

Latest revision as of 02:41, 1 June 2024

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].

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].

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 \newcommand{\V}{\operatorname{V}}
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-\newcommand{\Conv}{\mathop{\scalebox{1.1}{\raisebox{-0.08ex}{$\ast$}}}}%
+\newcommand{\mathds}{\mathbb}</math></div>
-\usepackage{pgfplots}
+Let <math>X</math>, <math>Y\sim\mathcal{N}(0,1,\mathbb{R}^d)</math>. Show the following.
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-\newcommand{\mathds}{\mathbb}</math></div>
-\label{PROB-2}Let <math>X</math>, <math>Y\sim\mathcal{N}(0,1,\mathbb{R}^d)</math>. Show the following.
 <ul style="list-style-type:lower-roman"><li> <math>\forall\:d\geqslant1\colon\E(\|X-Y\|-\sqrt{2d})\leqslant1/\sqrt{2d}</math>.
 </li>
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 </li>
 </ul>
-\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>.
-}