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

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.

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].

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

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 Figure and Table by running the corresponding experiments.