I have two independent unknown points $x, y \in \mathbb{R}^3$ and a set of $N$ observations $x_i$, $y_i$ of their positions that I model with a normal distribution: $x_i \sim \mathcal{N}(x, \Sigma_x)$, $y_i \sim \mathcal{N}(y, \Sigma_y)$, for each $i \in [1, N]$. Covariances $\Sigma_x$ and $\Sigma_y$ are known. I compute the distance between each pair of observations: $d_i = \|x_i - y_i\|$. I would like to test whether all the distances $d_i$ are similar, or if any of them is very different from the others. I have two questions:
If I model the distance between points $x$ and $y$ as a random variable $D$ (with samples $d_i$ or $d_i^2$), what distribution fits $D$?
I think I can test whether all the $d_i$ are similar with a chi-squared test. For that, I must have an expected variance for $D$, say $\sigma_d^2$, set the null hypothesis to $H_0: \sigma^2 \leq \sigma^2_d$, compute the test statistics $T$ variable and compare with the value of the $\chi^2$ for $N-1$ degrees of freedom. Is this correct, or is any other type of test best suited for this problem?