Most of these distributions happen because you have to limit or normalize the random number.
We normalize it to be all positive, fit within a range, and even to fit within the constraints of the memory size for the assigned variable type.
In other words, because we have to limit the random call between 0 and X (X being the size limit of our variable) we will have a group of "random" numbers between 0 and X.
Now when you add the random number to another random number the sum will be somewhere between 0 and 2X...this skews the values away from the edge points (the probability of adding two small numbers together and two big numbers together is very small when you have two random numbers over a large range).
Think of the case where you had a number that is close to zero and you add it with another random number it will certainly get bigger and away from 0 (this will be true of large numbers as well as it is unlikely to have two large numbers (numbers close to X) returned by the Random function twice.
Now if you were to setup the random method with negative numbers and positive numbers (spanning equally across the zero axis) this would no longer be the case.
Say for instance RandomReal({-x, x}, 50000, .01) then you would get an even distribution of numbers on the negative a positive side and if you were to add the random numbers together they would maintain their "randomness".
Now I'm not sure what would happen with the Random() * Random() with the negative to positive span...that would be an interesting graph to see...but I have to get back to writing code now. :-P