Let $\mathbf{y_1} =\begin{bmatrix}g_1x_1 & g_2x_1 & \dots & g_Nx_1 \end{bmatrix}$ and $\mathbf{y_2} = \begin{bmatrix} f_1x_2 & f_2x_2 & \dots & f_Nx_2\end{bmatrix}$. All the elements of $\mathbf{g}=\begin{bmatrix} g_1&g_2 &\dots &g_N\end{bmatrix}$ and $\mathbf{f}= \begin{bmatrix} f_1 & f_2 & \dots f_N\end{bmatrix}$ 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 $\mathbf{y_2}$?


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