1.

It is clear from the definition of a marginal pmf that they would be the same, given you are summing either from 0 to infinity of i, or j to calculate either marginal.

2.

Letting

, to calculate

we note;

For:

Where by the definition of factorials

Since

So iterating for all k, we end up with the given formula.

3.

For any pmf, we know that the sum over all possible values equals 1. So we can solve for

Thefore,

, so S is poison distributed with paramater

4.

By the definition of independence;

LHS:

RHS:

as X and Y have the same distribution.

Hence they are not independent

5.

Since

its expected value is

Using the fact from (a), X and Y have the same distribution and hence same expected value;

Therefore,