Expected distance between (X, Y), where both X and Y are standrd normal random variabls and the origin

Question

Let $(X, Y)$ be two independent standard random variable, with mean and SD being 0 and 1 respectively.

What would be $E[\sqrt(X^2 + Y^2)]$, the expected distance between $(X, Y)$ and the origin.

This reads like a textbook style question. Is this for some subject? — Glen_b, Nov 12 '19 at 08:35
However, it's answered a few times on site already - sometimes in a more general form (e.g. see [here](https://stats.stackexchange.com/questions/272885/intuition-for-rayleigh-pdf/272932#272932)) — Glen_b, Nov 12 '19 at 08:42

score 2 · Answer 1 · answered Nov 12 '19 at 08:35

2

$Z=\sqrt{X^2+Y^2}$ is a Chi distributed random variable with degree of freedom $k=2$. Its mean is $$\mu=\sqrt{2}\frac{\Gamma((k+1)/2)}{\Gamma(k/2)}=\sqrt{2} \Gamma(3/2)=\sqrt{\frac{\pi}{2}}$$

answered Nov 12 '19 at 08:35

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Expected distance between (X, Y), where both X and Y are standrd normal random variabls and the origin

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