Average minimum distance between two random vectors - Cross Validated most recent 30 from stats.stackexchange.com 2019-07-16T06:50:38Z https://stats.stackexchange.com/feeds/question/93866 http://www.creativecommons.org/licenses/by-sa/3.0/rdf https://stats.stackexchange.com/q/93866 1 Average minimum distance between two random vectors user4259 https://stats.stackexchange.com/users/43826 2014-04-15T14:00:15Z 2014-04-15T14:00:15Z <p>Let $\mathbf{y_1} =\begin{bmatrix}g_1x_1 &amp; g_2x_1 &amp; \dots &amp; g_Nx_1 \end{bmatrix}$ and $\mathbf{y_2} = \begin{bmatrix} f_1x_2 &amp; f_2x_2 &amp; \dots &amp; f_Nx_2\end{bmatrix}$. All the elements of $\mathbf{g}=\begin{bmatrix} g_1&amp;g_2 &amp;\dots &amp;g_N\end{bmatrix}$ and $\mathbf{f}= \begin{bmatrix} f_1 &amp; f_2 &amp; \dots f_N\end{bmatrix}$ are drawn from a rayleigh distribution. $x_1$ and $x_2$ are taken randomly from the set $\{-1,1\}$ with equal probability, i.e, $x_1$ can be $1$ or $-1$ with $0.5$ probability. There are four combinations for $(x_1, x_2)$: $(1,1),(1,-1),(-1,1),(-1,-1)$ which will lead to four combinations for $(\mathbf{y_1}, \mathbf{y_2})$ as well. Considering all combinations, how can I find the average $\it{minimum}$ euclidean distance between the vectors $\mathbf{y_1}$ and $\mathbf{y_2}$?</p>