# Proving that two variables are antithetic variables

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.

• Start at stats.stackexchange.com/questions/28992. To appreciate the need for condition (2), consider a random variable $X$ that takes on the values $\pm 1$ with equal probability, the value $0$ with positive probability, and has no other possible values. – whuber Jun 16 '18 at 11:52
• @Martin when you don't know where to start it often means that you don't have all the information you need in your head. What is the definition of antithetic variables that you were given? – Glen_b -Reinstate Monica Jun 17 '18 at 0:17