I have a normally distributed random variable $X$. I sample two points $x_1$ and $x_2$, and I am interested in the absolute difference between these two sampled points: $d=|x_2-x_1|$.
I repeat this $N$ times collecting $N$ pairs of points, giving me absolute differences of $d_1, d_2, ... d_N$.
Is it possible to say anything about the distribution of $d$? Specifically, I would like to construct a confidence interval on $d$, i.e. "with $1-\alpha$ confidence, $d$ falls within this interval."