# how to compute the probability ditsribution of the average of n iid random variables

It has been a long time since I took a probability class, and I'm sure that this site is for any level of mathematics...

Given $n$ iid random variables, I want to compute the pdf of their average. How can I do that? I know I can do for summation but is it still true for the average? Can you give me a reference textbook for computing average, not just summation?

A related but not important question: I've read some papers on Monte Carlo simulation, but mostly they just focus on some tail bound but are not interested in actually computing the pdf. Is it because just bounding upper tail value is enough or is it because it's rather cumbersome to compute the exact of pdf?

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The average is just the sum divided by a constant. If you know the distribution function of the sum, the one for the average is easy. The probability distribution of the sum is not necessarily easy to find. That's why, in favourable cases, it is very helpful that by the Central Limit Theorem, the sum (and therefore the average) has a distribution which can be well approximated by the normal, if $n$ is large. – André Nicolas Nov 22 '11 at 4:56
@André Nicolas Thanks! – Endo Nov 23 '11 at 16:06