My question is actually quite short, but I'll have to start by describing the context since I am not sure how to directly ask it.
Consider the following "game":
We have a segment of length n ("large segment") and m integers ("lengths"), all considerably smaller than n. For each of the m lengths we draw a random sub-segment of its length on the large segment. For example, if the large segment is of size 1000 (i.e. 1..1000) and we are given lengths 20, 10, 50, than a possible solution would be: 31..50, 35..44, 921..970 (sub-segments of lengths 20, 10 and 50 respectively).
Notes: 1. This is just a toy example. We usually have many more lengths so there are many overlaps and each position in the large segment is covered by multiple sub-segments. 2. Remember that the lengths are given; only their mapping to the large segment is random. 3. Drawing a sub-segment of length k is done bu simply drawing a number from a uniform distribution over 1..n-k (a sub-segment of size k can start at position 1, 2, ... n-k).
Now, we conduct many simulations of the process an d record the data. We finally examine for each position the distribution of number of sub-segments covering this position. If we look at positions that are relatively far from the edges of the large segment, the distribution in each such position is normal, and all the distributions look the same. The "problem" is that the positions at the ends do not look normal at all. This is not surprising, since, for example, if we are now drawing a sub-segment of length 10, the only way the very first position in the large segment will be covered is if we draw 1, whereas, for example, the 10th position will be covered if we draw 1,2,3,..10.
What I am trying to figure out is what is the kind of distribution we see in the "edge" positions (it's not normal, but I think it usually looks like a normal distribution with its tail cut in one direction), and also how can I approximate this distribution density function from my simulations. For the "center" positions, I just estimate the mean and standard deviation and since I beleive the distributions there are normal - I can use the normal density function. This alos makes me think if I really need to treat the positions in a categorical way - "near the edges" and "not near the edges", or whether there are actually the same in some sort (some generalization of the normal distribution?).
Thank you, and sorry again for the length of the post.