MathDB

Let

\mathcal{N}_p

stand for a

p

dimensional random variable of standard normal distribution. For

a\in\mathbb{R}^p

, let

H_p(a)

stand for the expectation

E|\mathcal{N}_p+a|

. For

p>1

, prove that

H_p(a)=(p-1)\int_0^{\infty} H_1\left( \frac{|a|}{\sqrt{r^2+1}}\right) \frac{r^{p-2}}{\sqrt{(r^2+1)^p}} \mathrm{d}r

Problem Statement