Let x be a random variable that takes integer values and is symmetric, that is, p(x = k) = p(x = âk) for all integers k. what is the expected value of y = cos(xï) and y = sin(xï)?

Without knowing how is distributed, or what its PMF happens to be, we can't completely answer the question. However, recall that




If , then



By the symmetry of ,



Splitting on the even and odd , and using the fact that for , we can write this as



but without knowing how is distributed, we can't go any further as far as I'm aware.

For , we would have



but for all integers . Thus .