Approximating density function for a non-normal distribution 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.
 A: I'm not sure I understand your question exactly, but I assume you are looking for the probability mass at each point, where an event is defined as a subsegment covering a particular position.  If this is true, I believe you should be able to work out the exact probability mass function.
For each subsegment of length K the distribution at a particular location between K and N-K+1 is uniform and proportional to K.  The distribution at the tails is stepwise increasing from 1 to K.  This can be seen by just working out one example.
Given multiple subsegments of different sizes, simply add these functions up.  You can then normalize everything so the weights sum to one if you want a proper distribution function.  Given a length and a list of subsegment lengths, this probability mass function should be easy to code up in the language of your choice.
If you are interested in the number of segments covering any point, this is simply the sum of Bernoulli random variables with different p, defined at each point using the same pmf.  The simplest approach would be to enumerate the possibilities from 1 to K.
