Multiplying values from a uniform distribution approximates to which distribution?

Question

I would like to informally explain the relationships among uniform, normal and exponential distributions by stating that:

1. the sum of values obtained by sampling a uniform distribution approximate to a normal distribution. Example: if I keep collecting the sums of three dice, I will get a symmetric distribution.

2. the multiplication of values obtained by sampling from a uniform distribution approximate to an exponential distribution. Example: if I keep collecting the products of three dice, I will get a distribution which is very skewed to the right.

library(extraDistr)

set.seed(0)

uniforme <- rdunif(n = 1000, 1, 6)
barplot(table(uniforme), main = "Results of a die")

normale <- rdunif(n = 1000, 1, 6) + rdunif(n = 1000, 1, 6) + rdunif(n = 1000, 1, 6)
plot(density(normale), main = "Sums of three dice")

exponentielle <- rdunif(n = 1000, 1, 6) * rdunif(n = 1000, 1, 6) * rdunif(n = 1000, 1, 6)
plot(density(exponentielle), main = "Products of three dice")

Is this roughly correct? In case it is incorrect to say "exponential distribution" in this case, what would be the correct name of the resulting distribution?

Answer 1

The first statement (sum of uniform approximating a normal) is not incorrect, but is an understatement:

• It in fact is the sum of any identically distributed random variables which approximates a normal, not just uniformly distributed. And that is because of the CLT (Central Limit Theorem), which is a much more general property than your first statement: the sum of n identically random variables is normally distributed.
• Now, this property is asymptotic, that is is correct if n is "large enough". For the uniform distribution (which is quite well behaved), n indeed does not need to be too large. With summing 3 dies, you are close to normal, but not quite there; for 5 you are "pretty darn" close (close enough for all practical purposes). But for other distributions, you would need much larger n's (10's, 100's, or more for extremely skewed distributions).
• In fact, even the sum of non-identically distributed random variables will asymptotically be normal (but with larger n's; but sutrprisingly not always...)
• This is a pretty nice blog, with animated simulations, and lots of interesting graphs, which illustrates all this.

The 2nd statement is unfortunately not correct.

• You can find nice derivations of the PDF of the product of n uniform random variables here or here (I like the derivation in the 2nd a bit better, fwiw).
• Now, as @Henry commented, if you take the log of this product, it turns into a sum (of the logs of the individual variables), and we know what happens to that sum of i.d. random variables. Asymptotically, it becomes a normal distribution, because of the CLT. So the product becomes lognormal.
• But that is not specific to the uniform distribution; it would be true for any product of i.d. variables, and in fact for any product of non i.d. variables. And that would be just the result of applying one of the key properties of the log, and then using the CLT.

So there is no particular relationship among uniform, normal and exponential distributions; but there is (an obvious) one between normal and lognormal. And another (not so obvious one) between any distribution(s), and the normal.