I have implemented what I believe to be a sampler from a distribution over the integers. Its a specific discretization of the normal distribution so I know stuff like mean and variance of the theoretical distribution, and I can compute an approximate of the theoretical pmf but it's expensive so I can't do it for a lot of values. I'd like to gain some confidence that my sampler is actually sampling from the right distribution, preferably in a reproducible way (so no looking at histograms etc). what's a reasonable test for this?
I'm mostly overwhelmed by the variety of available statistics and the search for one that is defined for discrete distributions yet tells me something about "normality".