I have read these two Q&A:

PDF of the product of two independent Gamma random variables

How to Plot the PDF of Product of two Normals

I have two correlated Meixner random variables, $X$, and $Y$, where

$\qquad X\sim MXN(a=0.03306, b=0.30800, m=-0.00099, d=0.44168)$, $\qquad Y \sim MXN(a=0.03064, b=0.45599, m=-0.00173, d=0.51881)$

I need compute the product distribution the $Z=X \cdot Y$. The joint distribution of $X$ and $Y$ is the Student's $t$ copula model with two parametrs: correlation $\rho=0.722$ and degree of freedom $v=7.566$.

I have tried

Z = 
    {x \[Distributed] MeixnerDistribution[0.03306, 0.30800, -0.00099, 0.44168], 
     y \[Distributed] MeixnerDistribution[0.03064, 0.45599, -0.00173, 0.51881]}]

and TransformedDistribution gives Mean[Z]=2.50021*10^-6 output only.


I have tried Plot[PDF[Z, x], {x, 0, 1}] but the Mathematica software v.10 is running. I'm looking for the theoretical solution.


How to compute the product distribution of two Meixner variables?

  • 1
    $\begingroup$ Hmm, I believe the first parameter of MeixnerDistribution[] ought to be positive. Maybe check if the definition you're using matches Mathematica's? $\endgroup$
    – J. M.'s torpor
    Mar 19 '18 at 4:38
  • $\begingroup$ In support of @J.M.'s comment: MeixnerDistribution >> Details: MeixnerDistribution[a,b,m,d] allows m to be any real number, a and d to be any positive real number, and b such that -π <b<π. $\endgroup$
    – kglr
    Mar 19 '18 at 5:00
  • $\begingroup$ @kglr, I have edited the order of arguments in MeixnerDistribution[]. $\endgroup$
    – Nick
    Mar 19 '18 at 6:16
  • $\begingroup$ Version 11.3 fails with PDF[Z, x]. $\endgroup$
    – user64494
    Mar 19 '18 at 6:18
  • 1
    $\begingroup$ Up to Wiki (see en.wikipedia.org/wiki/Kendall_rank_correlation_coefficient ), the Kendall rank correlation coefficient is not it. The correlation (see en.wikipedia.org/wiki/Correlation_and_dependence ) is required. $\endgroup$
    – user64494
    Mar 19 '18 at 7:36

You can obtain an approximate plot of PDF[Z, x] in such a way.

Z = TransformedDistribution[x*y, {x \[Distributed]
MeixnerDistribution[0.03306, 0.30800, -0.00099, 0.44168], 
y \[Distributed] MeixnerDistribution[0.03064, 0.45599, -0.00173, 0.51881]}];
RandomVariate[Z, 10^4];Histogram[%, Automatic, "Probability"]

enter image description here

Addition. The answer which is valid for independent random variables was submitted before the edit of the question.

  • $\begingroup$ This is a good answer. "The next best thing to having an explicit formula for the probability density function is having a gazillion random samples from that distribution." (Are you going to modify to account for the OP's edit and maybe use SmoothKernelDistribution rather than a histogram?) $\endgroup$
    – JimB
    Mar 19 '18 at 14:22

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