Question

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)?

Answer 1

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

$\cos(-kx)=\cos kx$
$\sin(-kx)=-\sin kx$

If $Y=\cos(\pi X)$, then

$\mathbb E[Y]=\displaystyle\sum_{k\in\mathbb Z}\cos(k\pi)\,\mathbb P(X=k)$

By the symmetry of $X$,

$\mathbb E[Y]=\mathbb P(X=0)+2\displaystyle\sum_{k\ge1}\cos(k\pi)\,\mathbb P(X=k)$

Splitting on the even and odd $k$, and using the fact that $\cos(k\pi)=(-1)^k$ for $k\in\mathbb Z$, we can write this as

$\mathbb E[Y]=\mathbb P(X=0)+2\displaystyle\sum_{\ell\ge1}\bigg(\mathbb P(X=2\ell)-\mathbb P(X=2\ell-1)\bigg)$

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

For $Y=\sin(\pi X)$, we would have

$\mathbb E[Y]=\mathbb E[\sin(\pi X)]=\displaystyle\sum_{k\in\mathbb Z}\sin(k\pi)\,\mathbb P(X=k)$

but $\sin(k\pi)=0$ for all integers $k$. Thus $\mathbb E[\sin(X\pi)]=0$.