# How can I derive the product distribution of two Meixner random variables?

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 =
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]}]


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

Edit

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.

Question

How to compute the product distribution of two Meixner variables?

• Hmm, I believe the first parameter of MeixnerDistribution[] ought to be positive. Maybe check if the definition you're using matches Mathematica's? – J. M.'s ennui Mar 19 '18 at 4:38
• 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. – kglr Mar 19 '18 at 5:00
• @kglr, I have edited the order of arguments in MeixnerDistribution[]. – Nick Mar 19 '18 at 6:16
• Version 11.3 fails with PDF[Z, x]. – user64494 Mar 19 '18 at 6:18
• 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. – 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]

• 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?) – JimB Mar 19 '18 at 14:22