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I've been reading a text on probability theory and it thoroughly looked at the distribution of the sum of i.i.d. random variables but on the last page, they made a comment about the distribution of the product of random variables that confused me:

They said that if you had i.i.d. random variables $Y_1,Y_2,...$ that are uniformly distributed on (0,1), then the distribution of $Y_1Y_2\cdots Y_n$ is approximately $e^{X}$, where X is normal with mean $n\mu$ and standard deviation $\sigma\sqrt{n}$ (for some $\sigma>0$ and real $\mu$. Can someone please explain why this is?

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Write the product as the exponential of a sum of i.i.d. random variables and apply the central limit theorem to these. – Did Nov 14 '11 at 21:17
In other words, apply the central limit theorem to the logarithm of the product. – Michael Hardy Nov 14 '11 at 21:37
Let $W_i=\log(Y_i)$. In principle you even almost don't need to compute the mean and variance of $W_i$, since you are not asked to identify $\mu$ and $\sigma$. (But you do need to know the mean and variance exist.) However, the calculation of the mean and variance of $\log(Y_i)$ is not hard by integration by parts. – André Nicolas Nov 15 '11 at 0:16

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