I have two probability distributions $p(x) = N_1 x^{-\tau} \exp(-\frac{x}{x_0})$ and $p(y) = N_2 y^{-\kappa} \exp(-\frac{y}{y_0})$. $N_1$ and $N_2$ are just normalization constants and $x>0$, $y>0$, $x_0>0$, $y_0>0$, $\tau>0$, and $\kappa>0$. I want to know what is the distribution of x*y. I am not interested in the normalization constant.

I tried many things via characteristic functions or convolution, but could not find any analytic solution. I suspect there is no analytic solution, because of the nasty integral involved. Thus, I want to know if there is a clever approximation. For example, prove that the new distribution is again a power law for small x*y, and the exponential cut-off on the new distribution is $>x_0$ and $>y_0$. Empirically, I find if I take the product of multiple $x$, let's say $m$-times, with distribution $p(x)$ as above, the resulting distribution approaches a power law. Can this be proven for the limit $m\rightarrow \infty$?

  • $\begingroup$ Shouldn't that be $x^{-\tau}e^{-x_0/x}$? The functions you've written down aren't actually integrable at small $x$. $\endgroup$ – userN Oct 18 '12 at 3:35

So your random variables $X$ and $Y$ have Gamma distributions with scale parameters $x_0$ and $y_0$ and shape parameters $1-\tau$ and $1-\kappa$ respectively (of course, you must assume $\tau < 1$ and $\kappa < 1$ for these to be well-defined). I presume you are also assuming $X$ and $Y$ are independent. Maple says the probability density function of $Z=XY$ is $$ f_Z(z) = \frac{2}{\Gamma(\tau) \Gamma(\kappa)} {z}^{{\tau/2}+{\kappa/2}-1}{y_{{0}}}^{-{\tau/2}-{\kappa/2}}{x_{{0}}}^{-{\tau/2}-{\kappa/2}} {{\rm K}_{\kappa-\tau}\left({\frac {2 \sqrt {z}}{\sqrt {x_{{0}}y_{{0}}}}}\right)} $$ for $z > 0$, where $K_{\kappa-\tau}$ is a modified Bessel function of the second kind.

  • $\begingroup$ Thanks. I also came across the modified Bessel function of the second kind. Is there a good approximation of K? $\endgroup$ – Heiko Hoffmann Oct 18 '12 at 0:56
  • $\begingroup$ For example, series expansion around $0$: for non-integer $\nu$, $$K_\nu(x) = \sum _{k=0}^{\infty }{\frac {{2}^{\nu-2\,k-1}\pi \,{x}^{-\nu+2\,k}} { \left( k-\nu \right) !\,k!\,\sin \left( \pi \,\nu \right) }}-{ \frac {{2}^{-\nu-2\,k-1}\pi \,{x}^{\nu+2\,k}}{ \left( \nu+k \right) !\,k !\,\sin \left( \pi \,\nu \right) }} $$ See also wolframalpha.com/input/?i=BesselK[nu%2Cx] $\endgroup$ – Robert Israel Oct 18 '12 at 2:40

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.