The real relationship is between the geometric and the Neg Bi.

The rv in the geo is the trial in which the first success appears.

The rv in the neg bi is the trial in which the success appears.

So, if r=1, the neg bi becomes the geo.

And the geo is a sum of r i.i.d. geo's.

Both of these rvs have infinite support.

BUT with the Binomial there is a fixed number of trials.

In the other two we can keep waiting for Godot, which may never happen.

I work with the St petersburg game and thats an example where we wait a long time for a big pay out.

It has infinite expectation, which confused the Bernoulli's 295 years ago.

BUT my solution to that game does work.