Replicate the results of Figure and Table by running the corresponding experiments.

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].

Replicate the results of Table. Make additionally a plot of the distribution of mutual distances (this should give a picture similar to Figure).

Replicate the results of Table. Let your code also compute the averages and variances of scalar products of the normalized vectors, i.e., [math]\langle{}x/\|x\|,y/\|y\|\rangle{}[/math].

Compute in a simulation norm, distance and scalar product of points that are drawn from the hypercube [math]H_d[/math] (coordinate-wise) uniformly, i.e., [math]x=(x_1,\dots,x_d)[/math] is drawn such that [math]x_i\sim\mathcal{U}([-1,1])[/math] for [math]i=1,\dots,d[/math]. Make plots and tables similar to Figure and [[#TAB-1 |Table\,--\,]]. Compare the experimental data with our theoretical results above.

If you enjoy horror movies, then watch 2002's Hypercube.