For your first question..
I haven't answered your second question but if you're trying to find E(X) for a geometric distribution I think there is an easier way.
In a derivation for the expectation of an RV with a geometric distribution I have the following:
I have several questions.
1. What sorcery took place with the binomial coefficient and k, such that k disappeared, r appeared as a factor and its now rather than ?
2. Presumably the logic is that . Can anyone show me how to prove this and what class of distribution is this, not binomial right?
Many thanks in advance. MD.