Given the joint probability distribution for three dependent continuous random variables, I want to find the probability distribution for the product of these random variables.

I found this relevant question for two dependent variables.

I also found a question that deals with the case of three variables, but only in the special case where the variables are independent.

But what is the answer in case of three dependent continuous random variables?

Let's consider the distribution $E$ with $e=abc$, with $A$, $B$, $C$ being three dependent continuous random variables. Can we say that:

$$ f_E(e) =∫^\infty_{−\infty}∫^\infty_{−\infty}f_{A,B,C}\left(a, b,\frac{e}{ab}\right) \space \frac{1}{|ab|} \space da \space db ?$$

  • $\begingroup$ I edited your post to give emphasis to what you're asking, and to highlight why the second link doesn't fully address your question. Please check this edit to ensure that it preserves your intent. $\endgroup$
    – Kenny Wong
    May 16 at 12:26

1 Answer 1


First, construct the cdf, the $P(x_1x_2x_3 < x)$ with the triple integral $$\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \int_{-\infty}^{x/(x_2x_3)} J(x_1,x_2,x_3) dx_1dx_2dx_3$$ Then you can differentiate the cdf with respect to x to get the pdf.

  • $\begingroup$ Thanks for the reply ! Does it mean that the formula that I gave above is wrong ? $\endgroup$
    – MadMax2048
    May 16 at 13:08
  • $\begingroup$ I think it’s equivalent to what I have above, with the first integral calculated, and the differentiation by x done under the integral. $\endgroup$
    – Paul
    May 16 at 17:08
  • $\begingroup$ Paul think it is equivalent but it seems that a doubt remain. Someone to confirm if my formula is correct ? $\endgroup$
    – MadMax2048
    May 17 at 14:46
  • $\begingroup$ Performing numerical simulations seems to validate the formula. However I would have prefer to get a mathematical demonstration of its validity. $\endgroup$
    – MadMax2048
    May 26 at 14:10

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