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I would like to know why this equality $f_x,_y (x, y; θ) = f_xy (x; θ) × f_y(y; θ)$ holds.

so, why the joint density of two random variables is equal to the conditional density of the first given the second times the second?

It's a given axiom in every book I consulted, but I would like the proof.

MaxSim423
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1 Answers1

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This property of probability densities is based on the so-called Law of Total Probability. The proof can be found here: https://proofwiki.org/wiki/Total_Probability_Theorem

The Law of Total Probability can be applied to densities, since densities give the likelihood of an event occurring.

Red
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