# Estimating the Distribution of a Very Large Population of Known Size and Unknown Variance

I would like to estimate the distribution of a very large population of known size but unknown mean and variance. I cannot assume anything about the underlying distribution. The values of observations are always discrete. I cannot sample the entire population but I would like to estimate the probability density function. I would also like some level of assurance of the correctness of the estimated distribution. What is the best way to go about this?

• What do you mean by the statement "The values of observations are always discrete"? – Ashok Jun 20 '11 at 14:39
• @Ashok: In this case, I am certain that they are non-negative, non-zero integers. In general what I meant was that I believe that the distribution is lumped in some way rather than being continuously distributed over the entire range. – Misha Jun 20 '11 at 17:04

First, I assume that what you want to do is to estimate the distribution of some property of the population, no? One way to go about it would be to construct the empirical distribution function: $\mathbb{F}(x) = \frac{1}{n}\sum _{i=1} ^{n} I(X_i \leq x)$ ($n$ the number of observations). Once you have done this, you can use various test to compare this to any distribution function that you'd like (i.e., one that you suspect produced the data).