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

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 8:35
• However, it's answered a few times on site already - sometimes in a more general form (e.g. see here) – Glen_b Nov 12 '19 at 8:42

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