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To generate a sample from a one dimensional PDF, I know that you can invert the cdf (assuming it's invertible) and use a uniformly distributed random number plugged into that function to generate a number from the original PDF.

Is it possible to invert the CDF of a two dimensional PDF in the form $z=f(x,y)$?

I'm a little stumped how you'd turn such a PDF into a CDF, but am more stumped how you would invert it since it isn't just a simple matter of flipping the input and output variable - there being two inputs and one output.

Alan Wolfe
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    While it does not directly deal with inverting a 2D CDF, an approach in spirit of your question is in https://arxiv.org/pdf/1307.1223.pdf – Just_to_Answer Aug 05 '17 at 18:37

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