# Changing the basis of random variables

Let $X_1$ and $X_2$ be two independent (1-dimensional) random variables and let $Y_1 = f^1(X_1, X_2)$ ($f^1$ is a deterministic function) be a (1-dimensional) random variable too.

Question

Does there exist a random variable $f^2(X_1, X_2)$ which is independent of $f^1$ and such that $X_1 = g_1(f^1, f^2)$ and $X_2 = g_2(f^1, f^2)$ for some $g_1$ and $g_2$?

If yes, then $f^1, f^2$ can be viewed as a 'new basis' for $X_1, X_2$.

• Are you sure about what you're asking? Some of your conditions are contradictory. Since $f^1$, $f^2$, and $g_1$(you didn't say, but I take it that's what you want) are deterministic functions, your condition says $X_1 = g_1(f^1, f^2) = g_1( f^1(X_1, X_2), f^2(X_1, X_2))$, then $X_1$ cannot be independent of $X_2$, since it's a deterministic function of $X_2$. – horaceT Jul 4 '16 at 6:10
• @horaceT The flaw in your reasoning becomes apparent when you consider the case $f^1(X_1,X_2)=X_1$ and $f^2(X_1,X_2)=X_2$. – whuber Jul 4 '16 at 16:13
• Similar question - mathoverflow.net/questions/207664/… – Vivek Bagaria Jul 30 '16 at 0:26