# What does an infinite sum of uniform random variables yield?

19
Hey everyone.

I haven't taken statistics yet, but as a matter of interest I was contemplating the fact that uniform random variables added together seem to generate "bell curve" like distributions.

My question is if I add up an infinite number of equally distributed random variables will the resulting values be the normal/Gaussian distribution?

Or, perhaps will the effect of adding so many variables just cause the peak of the distribution to become excessively amplified until the function is just like one sudden spike at the center of the distribution?

What's the real behavior?

Furthermore, since brown noise is a sum of deviations from the current position (i.e. random variables) I wonder if brown noise could be considered to be normal/Gaussian noise.

Also, I've read online about something called "white Gaussian noise". What is it? How does it differ from non-white Gaussian noise? Is brown noise "Gaussian noise"? Is noise generated directly from a Gaussian curve (rather than from a random walk) what they call "white Gaussian noise"?

Last edited: Mar 14, 2012

19
Oh, and I should also say that I would average the infinite sum of random variables so as to reform it back to the original range.

In other words, if I added 50 random variables together as one value, then I would divide the sum by 50 to bring everything back to [0,1).

Would doing this at the limit as N approaches infinity give me the normal curve?

Also, would I need to shift the distribution on the x-axis for it to be the normal distribution? Why is the normal distribution I see in other sources centered on zero? Special reasons/properties?

3. ### chiro

Hey WraithGlade and welcome to the forums.

Your intuition is correct that the distributions will look more like a bell curve with more distributions, but unfortunately you can't add 'infinitely' many uniform distributions because you won't be able to describe the distribution with 'real' parameters.

The easiest example is the mean. The mean of a standard uniform distribution is 1/2 for U(0,1) which is considered the standard uniform in many circumstances. Therefore for the infinite sum, you will have an infinite mean and this doesn't make sense when you're not only trying to calculate the first moment, but also any of the other moments (like with variance).

But yes if you want to see it for yourself, get a statistical software package (R is a free one) can calculate say 10 or 20 independent uniform distributions for say 1000-10000 values each (which will only take a short while) and then look at the histogram for the sum. In a past undergraduate class I had to do this exact same exercise.

For brown and white-noise, it would help if you gave a link to the definition. I don't want to give a perspective on something that is not relevant to this discussion.

As for the second question, the idea of 'normalizing' the distribution is pretty much what is involved in what is known as the Central Limit Theorem:

http://en.wikipedia.org/wiki/Central_limit_theorem

Basically if you take the sum of random variables and divide the total by the number of random variables, then standardize the distribution in terms of the mean (i.e. make a mean of zero) and correct with the right variance then you'll get a standard normal distribution.

The wiki page gives the identity and if you are interested in the derivation, MGF's can be used to prove the result.