# What is the mean and variance of this probability distribution?

I don't know if this distribution has a name or not so the best I can do is describe how it is obtained. You start with two arrays of n uniform random variables U(0,1) where n>2. I will provide a numeric example for the case of n=5 to make it easier to explain; however, n can take any value above 2.

So, for the case of n=5, the random variables for the first array may take the values

[0.8147 0.9058 0.1270 0.9134 0.6324]

and for the second array,

[ 0.0975 0.2785 0.5469 0.9575 0.9649].

You then subtract the minimum value of each array from it's respective array and obtain for the first array

[0.6877 0.7788 0 0.7864 0.5054]

and for the second,

[0 0.1810 0.4493 0.8600 0.8673].

You then divide by the new maximum, yielding for the first array

[0.8746 0.9904 0 1.0000 0.6426]

and for the second,

[0 0.2086 0.5181 0.9915 1.0000].

From henceforth I will refer to the arrays as A and B respectively. You now perform the operation 1-abs(A-B) yielding

[0.1254 0.2182 0.4819 0.9915 0.6426].

You then calculate the mean of the above array, yielding the final variable for which the probability density function of interest is obtained. I will refer to this variable as Z. For this case Z=0.4919, however, Z can of course take on any value between 0 and 1 for any n>2.

I would like to know the mean and variance of Z as a function of n.

On a separate but related note, I think it's worth mentioning that when I simulate 1,000,000 trials of the process for obtaining Z for n=3 in MATLAB and plot a histogram of the values obtained for Z to visualize the distribution, I obtain a distribution that is the same form as the first figure shown in this Wikipedia article about the Central Limit Theorem:

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

The histogram I obtain is below

and the figure from the Wikipedia article is shown below as well

As you can see they are very similar in shape, and I doubt this is just a coincidence. Does anyone know if this particular pdf has a name that I could look up and do more research on? Sorry I know I deviated from my original question about the mean and variance as a function of n (which is still the primary thing I would like answered), but I couldn't resist pointing out the similarity between the two distributions.

• From that wiki article you can see the density is just one they have choosen arbitrarily as a starting point! So the similarity is indeed, just a coincidence. As for your quation, I cannot see a (fast) way to find the expectation and variance analytically. Most probably you must bwe content with approximations. – kjetil b halvorsen Nov 4 '14 at 14:31