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Mathematical theory of statistics, concerned with formal definitions and general results.
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How do I find the PDF from a multidimensional CDF with indicator functions?
When in doubt, deal with indicator functions (and other awkward creatures like absolute values) by breaking the problem into cases.
Here we have
$$F(x_1,x_2) =
\begin{cases}
(kx_1^2)(kx_2^2) & \text{ …