TITLE: How do you call the problem of approximating a continuous distribution with a simple discrete distribution?
QUESTION [5 upvotes]: The following problem came up on the Mathematica forum as "Generating a list of integers that roughly satisfy a distribution": Given $n$, find $n$ integers (possibly with duplicates) whose distribution approximates a given normal distribution.
For example, given a normal with mean 25 and standard deviation 7, the following multiset could be a decent candidate :
{11,15,17,18,20,21,22,23,24,25,25,26,27,28,29,30,32,33,35,39}

I understand that the term "approximates" is not properly defined and there may be several ways of fixing that. For example, one could specify that random samples taken from the discrete distribution should "look like" (again, need to specify what "looks like means") they are drawn from the continuous distribution (conditioning the normal distribution on integrality or after rounding).
Are there references on that problem?

REPLY [2 votes]: The literature on order statistics might suggest approximating $X$ with $\{E[X_{k:n}]:1\le k\le n\}$.
For your $X$ with $n=20$, this gives
{11.9, 15.1, 17.1, 18.6, 19.8, 20.9, 21.9, 22.8, 23.7, 24.6,
25.4, 26.3, 27.2, 28.1, 29.1, 30.2, 31.4, 32.9, 34.9, 38.1}
So in a sample of size 20 from $X$, 11.9 is the expectation for the smallest element, and 15.1 is the expectation for the second-smallest element.  Then rounding gives
{12, 15, 17, 19, 20, 21, 22, 23, 24, 25, 25, 26, 27, 28, 29, 30, 31, 33, 35, 38}