Exercises

Apr 21'25

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Look up the CLT, which was done here in moments, learn how the convergence can be improved, and write a brief account of that.

BBot

Apr 21'25

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Prove that the area of the unit sphere in [math]\mathbb R^N[/math] is given by

[[math]] A=\left(\frac{\pi}{2}\right)^{[N/2]}\frac{2^N}{(N-1)!!} [[/math]]

with our usual convention [math]N!!=(N-1)(N-3)(N-5)\ldots[/math] for double factorials.

BBot

Apr 21'25

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Establish the following integration formula over [math]S^{N-1}_\mathbb R\subset\mathbb R^N[/math], with respect to the normalized measure, valid for any exponents [math]p_i\in\mathbb N[/math],

[[math]] \int_{S^{N-1}_\mathbb R}|x_1^{p_1}\ldots x_N^{p_N}|\,dx=\left(\frac{2}{\pi}\right)^{\Sigma(p_1,\ldots,p_N)}\frac{(N-1)!!p_1!!\ldots p_N!!}{(N+\Sigma p_i-1)!!} [[/math]]

where [math]\Sigma=[odds/2][/math] if [math]N[/math] is odd and [math]\Sigma=[(odds+1)/2][/math] if [math]N[/math] is even, where “odds” denotes the number of odd numbers in the sequence [math]p_1,\ldots,p_N[/math].

BBot

Apr 21'25

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Compute the density of the hyperspherical law at [math]N=4[/math], that is, the law of one of the coordinates over the unit sphere [math]S^3_\mathbb R\subset\mathbb R^4[/math].