I know that for marginal distributions $f(x)$ with cumulative distribution $F(x)$, we derive $f(x) = d F(x)/dx$

My question: If given a simultaneous cumulative probability density $F(x_1,x_2,\cdots,x_n)$ how can one derive $f(x_1,x_2,\cdots,x_n)$?

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    $\begingroup$ As written in Wikipedia, the pdf is the $n$-th derivative of the cdf against all its components. $\endgroup$ – Xi'an Apr 20 '17 at 19:55

It's just as you'd expect: the $cdf$ is differentiated with repect to each of the random variables. That is for a $cdf$ of n random variables $x_1,...,x_n$: $$f(x_1,x_2,...,x_n) = \frac{∂^n}{∂x_1 ...∂x_n} F(x_1,x_2,...,x_n)$$

However this must assume that the $cdf$ is continuous and the partial derivatives exist.

Some nice reference are: here and here


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