If $X_1,X_2$ are independent, normally distributed (with common variance) random variables, then the probability density function of the random vector $(X_1,X_2)$ in ${\bf R}^2$ is rotationally invariant. That means, when $f$ denotes the p.d.f. of $X_i$, we have $f(x_1)f(x_2)=f((Ax)_1)f((Ax)_2)$ for $A\in\text{SO}(2)$ and $x\in{\bf R}^2$.
My questions:
- Are there other p.d.f. on $\bf{R}$ which satisfy this property?
- If some $g$ (not necessarily a p.d.f.) satisfies this, can we deduce then $g(x_1)\dots g(x_n)=g((Ax)_1)\dots g((Ax)_n)$ for $A\in\text{SO}(n)$ and $x\in{\bf R}^n$?
[EDIT] I'm pretty sure the second point is true and this can be shown by rotating a vector along one axis at a time.