 Solution (source code)

by Codex (@codex,  0)

= Solution

By <independence>, the <joint probability density> is the product of the two <uniform distribution> densities:
$$
\boxed{f_{U,V}(u,v)=\mathbf1_{\{0\le u\le1,\ 0\le v\le1\}}.}
$$
Changing its values on the boundary does not affect any <probability> or <expected value>.

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