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I choose random points $X,\,Y$ in $[0,\,1]^n$ (I assume all $2n$ Cartesian coordinates are $U(0,\,1)$ iids). What is the probability distribution of $\left\Vert X-Y\right\Vert _{2}^{2}$? Even the $n=1$ case requires some care, since if I first fix $X$ then $Y$ has a uniform distribution with extrema of opposite sign.

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    $\begingroup$ You ask about the sum of identically distributed independent variables (the $(X_i-Y_i)^2$, whose distribution is readily computed). The methods explained in the investigation at stats.stackexchange.com/a/43075 will therefore be applicable to solving this problem. As that thread indicates, the result will be messy: to describe it with a formula typically requires $n$ separate expressions. Thus, it would help for you to indicate in what form you are looking for an answer as well as to tell us whether approximate solutions would be acceptable in your application, whatever it might be. $\endgroup$
    – whuber
    Sep 6 '17 at 13:48
  • $\begingroup$ @whuber An approximate CDF will do. $\endgroup$
    – J.G.
    Sep 6 '17 at 14:05
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For an approximate cdf, one possibility is the saddlepoint approximation. For that the mgf (moment generating function), or its logarithm, the cgf (cumulant generating function) is needed. For background see How does saddlepoint approximation work?. So let $X_1, \dotsc, X_n, Y_1, \dotsc, Y_n$ be iid $\mathcal{U}(0,1)$. Then $D^2 = \sum_i (X_i-Y_i)^2$ have mgf $$ \DeclareMathOperator{\E}{\mathbb{E}} \DeclareMathOperator{\P}{\mathbb{P}} M_n(t)=M_{\sum_i (X_i-Y_i)^2}(t)=M_{(X_1-Y_1)^2}(t)^n=\left(\E e^{(X_1-Y_1)^2 t}\right)^n $$ so the cgf is $$ K_n(t)=n\cdot \log\left\{ \E e^{(X_1-Y_1)^2 t} \right\} $$ which in this case is defined for all real $t$.

Without going into details, and with some help from maple, we find that $$ \E e^{(X_1-Y_1)^2 t} = {\frac {\sqrt {\pi}{\rm erf} \left(\sqrt {-t}\right){t}^{2}+{{\rm e}^{ t}} \left( -t \right) ^{3/2}- \left( -t \right) ^{3/2}}{ \left( -t \right) ^{5/2}}} $$ or alternatively $$ \E e^{(X_1-Y_1)^2 t} = {\frac {-2\,i{{\rm e}^{t}}{\it dawson} \left( i\sqrt {-t} \right) {t}^ {2}+{{\rm e}^{t}} \left( -t \right) ^{3/2}- \left( -t \right) ^{3/2}}{ \left( -t \right) ^{5/2}}} $$ but before using this numerically, more work will be needed to find a useful form.

I will come back later to add the resulting approximation.

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