the sum of the expectations OF the n different Y's.
IS the parameter Y?
Or are the random variables
where independence is not needed between the Y's.
SOME geo's start at 0 some start at 1.
It depends on your definition.
You might count the first trial or not.
That r is my n, the number of geo's that you're adding.
NOTE that if they are indep, then the sum is a neg bio.
BUT even if they are dependent then the expected value of the sum is still the sum of the expectations.