Using joint probability mass functions (multiple parts)

This question has a lot to it, and I'm guessing it has to be done in order (if so, you can't skip ahead a part without completing the prior part). The question(s) are listed below word-for-word.

Let and have the following pmf:

for some .

- Explain without any calculation why and have the same marginal pmf. That is, why .
- Let . Show that for all . Note that and need not be independent (as will be discussed below).
- Conclude the value of and recognize the distribution of . What is the parameter of this distribution?
- Compute . Are and independent?
- Find , and conclude .
**HINT**: To find from you may use part (a) that says and have same distributions and therefore same... - Compute .
**HINT**: You can write it as

,

and think of the infinite expansion of . - Conclude .
**HINT**: You may use a symmetry argument.

This is a lot to do, but that's the question as a whole, and I'd rather not make multiple topics pertaining to the same joint pmf.