I have a task in a subject called "Monte-Carlo methods" with which I'm a bit stuck and therefore I'm asking for your help.
The task is as follows: Prove that $g(X)$ and $g(2\mu - X)$ are antithetic variables, in case when: 1) $X$ is with symmetric density function in respect to $\mu$
2) $g$ is monotonically decreasing function
Can you help me, how to solve this one? To be honest, I don't have much ideas even where to start from.