I have discrete data that comes from the distribution of a discrete random variable Y. The data appears to follow the normal distribution (ie if I make a kde plot it looks like a normal distribution). I've conducted Bayesian inference and assumed that the mean mu follows a normal prior. Thus, the posterior of mu also follows a normal distribution.
Now I'm interested in conducting predictions for future samples from the pdf of Y given the samples I've already seen, but I'm unsure how to proceed. It doesn't make sense to use a normal distribution for prediction because all samples can only take non-negative Interger values. To me it doesn't seem to make sense to use a Poisson because the variance doesn't equal the mean nor does it make sense to use a Binomial because the value that the sample can take is theoretically unbounded (but in practice it is often bounded by ~25).