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Suppose $X_i\sim f_{X_i}(x)$, with each $X_i$ independent and identically distributed for some arbitrary probability density function $f_{X_i}(x)$. Given this information, and this information only, define the new random variable:

$$Y=\prod_{i=1}^nX_i$$

What is the distribution of $Y$? Particularly, what is its probability density function? If each $X_i$ is a discrete random variable, what's the analog? If there's no easy way to obtain $f_Y(y)$, is there a way to get the moment generating function of $Y$ instead?

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  • $\begingroup$ It is quite a job. If the $X_i$ are positive a.s. then you can start finding the distribution of $\ln X_i$, then $\sum_{i=1}^n\ln X_i$ by means of convolution and then $e^{\sum_{i=1}^n\ln X_i}$. $\endgroup$ – drhab Apr 14 '18 at 18:31
  • $\begingroup$ If there was a general answer, there won't be papers devoted to finding distributions of product of random variables having specific distributions! $\endgroup$ – StubbornAtom Apr 14 '18 at 18:36

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