Let $X = (x_1, x_2)$ and $Y = (y_1, y_2)$ where the random variables $x_1$, $x_2$, $y_1$, $y_2$ are independent standard normal. What is the expected distance between $X$ and $Y$, i.e. what is $$D_2=E\left(\sqrt{(x_1 – y_1)^2 + (x_2 – y_2)^2}\right)\ ?$$
Does this value increase or decrease when the number $n$ of dimensions increases, that is when $X = (x_1, \cdots, x_n)$ and $Y = (y_1, \cdots, y_n)$ for some independent standard normal random variables $x_i$, $y_i$, and $n>2$? Does it converge when $n \rightarrow \infty$?
Note: I do not know if the resulting integrals are tractable (I would suspect that they are not).
Best Answer
If everything is independent, then this is no longer a two-point problem: the variables $x_i - y_i$ are independent $N(0,2)$. The expectation is not terribly hard to compute, see here. It is increasing (moreover, the distribution itself is increasing, in terms of stochastic order) and, as @Did commented, equivalent to $\sqrt{2n}$.