Random Convex Hull

You are given two increasing arrays of $N$ integers $x_1 \lt x_2 \lt \ldots \lt x_N$ and $y_1 \lt y_2 \lt \ldots \lt y_N$. For a permutation $\sigma_1,\sigma_2,\ldots,\sigma_N$ of integers $1,2,\ldots,N$, let's define $A(\sigma)$ in the following way: - Consider the points $(x_1, y_{\sigma_1}), (x_2, y_{\sigma_2}), \ldots, (x_N, y_{\sigma_N})$. Then $A(\sigma)$ equals to the area of the convex h