The major difficulty of this question comes from the fact that its statement is false.

If has parameter , then this is obvious by symmetry, but otherwise it has no reason to be true.

Consider a binomial variable of parameter . Then is binomial of parameter . And , , so if your result was true, we would have (applying it to X and Y): and . However, if the expected value is not an integer, these terms sum to one, hence they would have to be equal to 1/2 : . Consider for instance a binomial r.v. of parameter such that , then is usually different from 1/2...