Hi Daniel, sorry I missed that last notification on your post ;)

ECDF is the cumulative DISTRIBUTION function - I guess the density you mention refers to the PDF (here it is density).

The CDF exists for both, a discrete or continuous distribution, but as you pointed out correctly, the CDF might not be a smooth function (while eCDF seems to give you an interpolated smooth function), but rather have jumps. I believe this is what you observed in your experiment, as you have probably gotten the interpolated values.

E.g. check out the CDF of the Poisson (PMF) and exponential (PDF) distributions that show these two quite nicely.

Another note on this topic (ITT): http://www.columbia.edu/~ks20/4404-Sigman/4404-Notes-ITM.pdf - has a great example to generate a binomial RV under ITT (page 2) - this might help you for your experiment.

Have a great week mate!