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.
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$\begingroup$ This reads like a textbook style question. Is this for some subject? $\endgroup$ – Glen_b Nov 12 '19 at 8:35
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$\begingroup$ However, it's answered a few times on site already - sometimes in a more general form (e.g. see here) $\endgroup$ – Glen_b Nov 12 '19 at 8:42
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$Z=\sqrt{X^2+Y^2}$ is a Chi distributed random variable with degree of freedom $k=2$. Its mean is $$\mu=\sqrt{2}\frac{\Gamma((k+1)/2)}{\Gamma(k/2)}=\sqrt{2} \Gamma(3/2)=\sqrt{\frac{\pi}{2}}$$
