Since you say your distribution is discrete, if your samples are independent, you could try a multinomial distribution as a model
You can use either maximum likelihood or a Bayesian estimator.
You could also take the mean and standard deviation and use a Gaussian model with rounding. A goodness of fit test
would tell you how good the model is.
If your samples are independent noise generation is straightforward . A Bayesian model is different because you use another (prior) noise generator as an input to the fixed parameter noise generator.
If your samples are correlated you can use a Gibbs sampler.
You should perform an independence test on your data, of which a Google search will provide a number of candidates
You also might consider boot-strapping
It is also very likely that a physics based noise model exists for your device.