Is $f(X,Y)$ a symetric random variable?

Question

If $X,Y$ are two real symetric random variables and $f : \mathbb R^2 \longrightarrow \mathbb R$ is an odd (measurable) function, is $f(X,Y)$ symetric?

• If $X$ and $Y$ are independant then yes because forall $B \in \mathcal B(\mathbb R)$ we get $$ \mathbb P [f(X,Y) \in -B] = \mathbb P[(-X,-Y) \in f^{-1}(B)] $$ and the class of the borelian sets $C$ of $\mathbb R^2$ such that $$ \mathbb P[(-X,-Y) \in C] = \mathbb P[(X,Y) \in C] $$ is a sigma algebra containing the elementary rectangles.
• If $Y = X$ or more generally if $Y = g(X)$ with $g$ odd we can adapt the latter reasoning to see that $f(X,Y)$ is still symetric.

What about the other cases?

Answer 1

$(X,Y)$ itself may not be symmetric even if both $X$ and $Y$ are. For instance $\mathbb P(X=\pm1)=\frac12$ and $Y=\xi X$ where $\xi$ is independent of $X$ and such that $\mathbb P(\xi=1)=\frac34=1-\mathbb P(\xi=-1)$.