# Given a sample of random variables, and n, how do I find the ecdf of the sum of n Xs?

I can't fit X to a common distribution so currently I just have X ~ ecdf(sample_data).

How do I calculate the empirical distribution of sum(X1 + ... + Xn), given n? X1 to Xn are iid.

My sample size is around $6\times10^7$. $n$ needs to be around 10-20.

To give an example of what I am doing: Suppose X is the age of a person. If I put 10-20 people in a room, what is the distribution of the sum of ges in the room? I have lots of peoples ages, and no reason to fit the distribution of an age to any of the common distributions.

(previously on StackOverflow - suggested to move here).

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The term empirical distribution refers to the sample distribution for the discrete set of n observations. If you have n observed values Xi i=1,...n then the sum is just a single number. So what do you want from it? Maybe I could say its empirical distribution is a point mass at the observe point. But that can't possibly be what you want to know. – Michael Chernick May 18 '12 at 12:47
@MichaelChernick, wouldn't it make sense to use the empirical distribution of the $X_{i}$'s, assume iid, and then use that to estimate the distribution of the sum? Of course, in that case you're right - empirical distribution would be the wrong terminology. – Macro May 18 '12 at 12:54
I have way more than n data points. I need to know n for roughly n<= 20, I have about 6x10^7 data points. – cammil May 18 '12 at 13:06
Now I'm confused about what the data actually is @cammil - can you clarify by editing the question? – Macro May 18 '12 at 13:28
Cammil, a possible interpretation of the question is that you have a large batch of $N$ data values $(x_1,x_2,\ldots,x_N)$; you stipulate a smallish number $n \le N$; and you wish to determine (or estimate) the distribution of $(x_{i_1}+x_{i_2}+\cdots+x_{i_n})$ as the indexes $(i_1,i_2,\ldots,i_n)$ range over all $n$-element subsets of $1,2,\ldots,N$. Am I on the mark? – whuber May 18 '12 at 14:50
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