TITLE: Is the distribution of a product of M iid uniform random variables really Log Normal?
QUESTION [5 upvotes]: Conventional wisdom says yes or mostly.  But consider the following simple derivation:
Let $y = \prod_{i=1}^M x_i$ where $x_i\sim U(0,1)$.  Then from independence, $E[y] = 2^{-M}$.  
Now, if we let $z = \ln y = \sum_{i=0}^M \ln x_i$ we can use the central limit theorem to claim that $z\sim Norm(\mu=-M,\sigma^2 = M)$ where we get $\mu = M \int_0^1 \ln x dx = -M$ and $\sigma^2 = M\int_0^1 (\ln x + 1)^2 dx = M$ from the standard change-of-variable equations for $\ln x$ combined with the CLT.  This leads to a Log-Normal distribution of y as:
$$f_Y(y) = \frac{1}{y\sqrt{2\pi M}}e^{-\frac{(\ln y +M)^2}{2M}} $$
Great! We got a distribution for $y$.  But if you try and calculate the expectation now, you get $e^{\mu + \frac{\sigma^2}{2}} = e^{-M/2} \neq 2^{-M}$.  What's worse, if you look at the asymptotic behavior, $e^{-M/2} \gg 2^{-M}$ as $M\rightarrow \infty$.
This doesn't make sense to me.  How can $z$ be considered "normal" in the limit of large numbers if it leads to a divergent systematic deviation in the calculation of the expectation of $y$? Is this an artifact of the finite support in $x_i$? If so, how to take it into account? More importantly, if $y$ is not really Log-Normal, to what degree are calculations of other properties valid? Ultimately, I need to be able to predict the most probable value (mode) of $y$.  Can I quantify the extent to which $y$ is not quite Log Normal and present some kind of bounds on the accuracy of my calculations?

REPLY [2 votes]: As you defined $z$, it will always be negative, and thus cannot be Normal. You can find its density as follows:
If $U$ is a Uniform $(0,1)$ r.v. then 
$P[-\log(U) \leq x] = P[U \geq e^{-x}] = 1 - e^{-x}$ for all $x \in [0,\infty)$, so 
$-\log U$ is an exponential r.v. with $\lambda=1$. That means that 
$$
     -\sum_{i=1}^M \log(x_i)
\sim \text{Gamma}(M,1),
$$
i.e., the density of $-Z$ is 
$$
  f_{-Z}(z) 
= \frac{z^{M-1} e^{-z}}{(M-1)!}, \ \ z > 0.
$$
Then using a simple change of variable the density for $Z$ is 
$$
  f_Z(z) 
= \frac{(-z)^{M-1} e^z}{(M-1)!}, \ \ z \leq 0.
$$
I also don't see how you've used the CLT.