I am trying to find the resulting PDF , when two random functions are multiplied. First function obeys normal distribution and second function obeys cauchy distribution. Can anybody tell me how to multiply the PDF of both of these functions and get the resulting PDF analytically.


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    $\begingroup$ You seem to be conflating multiplying random variables (and finding the density of the product) with multiplying the pdfs themselves. Please be absolutely clear about which you mean, since they're quite different operations. $\endgroup$ – Glen_b Dec 11 '14 at 5:03
  • $\begingroup$ In addition to Glen's request that you clarify whether you are seeking the product of 2 random variables (or trivially two pds?), could you please also specify whether you are using standardised forms or general forms of your Cauchy and Normal random variables. e.g. $N(0,1)$ or $N(\mu, \sigma^2)$? $\endgroup$ – wolfies Dec 11 '14 at 5:31
  • $\begingroup$ I want to find the PDF of the resulting function actually and First function is not a standard normal distribution,i.e. it has mean and variance,N(μ,σ2) and second function is a general cauchy distribution i.e. t student distribution with one degree of freedom. I am using the PDFs of both the function and using the formula f_z (z)=(∫_(-∞)^∞▒〖〖[f〗_x (x)f_y (z/x)]〗)/|x| dx to evaluate the integral. But it is getting increasingly complex after every step. $\endgroup$ – Anjan Tripathi Dec 11 '14 at 7:46
  • $\begingroup$ $$ \int_{-\infty}^{ \infty } fx(x)* fy(z/x) *(1/ \left| x \right| )dx$$ $\endgroup$ – Anjan Tripathi Dec 12 '14 at 3:47
  • $\begingroup$ A closed-form solution is certainly possible for $N(0, \sigma^2)$. The $\mu \neq 0$ case may be more difficult. $\endgroup$ – wolfies Dec 12 '14 at 5:25

Try the formula for product of random variables.

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    $\begingroup$ I am following the same pattern but the definite integral is getting complex after every step. $\endgroup$ – Anjan Tripathi Dec 11 '14 at 7:33

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