What is the mean euclidean distane between two points on the plane which coordinates are normally distributed?
I'm assuming this would be
$\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }\frac{e^{-\frac{(\text{x1}-\mu )^2}{2 \sigma ^2}} e^{-\frac{(\text{x2}-\mu )^2}{2 \sigma ^2}} e^{-\frac{(\text{y1}-\mu )^2}{2 \sigma ^2}} e^{-\frac{(\text{y2}-\mu )^2}{2 \sigma ^2}} \sqrt{(\text{x1}-\text{x2})^2+(\text{y1}-\text{y2})^2}}{\left(\sqrt{ 2 \pi } \sigma \right) \left(\sqrt{2 \pi } \sigma \right) \left(\sqrt{2 \pi } \sigma \right) \left(\sqrt{2 \pi } \sigma \right)}d\text{y2}d\text{y1}d\text{x2}d\text{x1}$
Does this integral have a functional representation?
UPDATE: After doing some numerical exprimentaion I'm guessing that the answer is $ \sigma \sqrt{\pi } $. But I don't know how to solve this integral.