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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.

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    $\begingroup$ What is an s.n.d. r.v.? $\endgroup$ – zoli Mar 19 '18 at 10:15
  • $\begingroup$ @zoli standard normally distributed, I changed it in the post. $\endgroup$ – fweth Mar 19 '18 at 10:29

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