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10

Part of 2004 Miklós Schweitzer

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Miklós Schweitzer 2004, Problem 10

Source: Miklós Schweitzer 2004

7/30/2016
Let Np\mathcal{N}_p stand for a pp dimensional random variable of standard normal distribution. For a∈Rpa\in\mathbb{R}^p, let Hp(a)H_p(a) stand for the expectation E∣Np+a∣E|\mathcal{N}_p+a|. For p>1p>1, prove that Hp(a)=(p−1)∫0∞H1(∣a∣r2+1)rp−2(r2+1)pdrH_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
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