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.
Letting , to calculate we note;
Where by the definition of factorials
So iterating for all k, we end up with the given formula.
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
By the definition of independence;
RHS: as X and Y have the same distribution.
Hence they are not independent
Since its expected value is
Using the fact from (a), X and Y have the same distribution and hence same expected value;