# Functions of three mutually independent random variables

Let $X$, $Y$, and $Z$ be mutually independent random variables. Let $h(X)$ be a function of $X$ only and $g(Y, Z)$ be a function of only $Y$ and $Z$. Are $h(X)$ and $g(Y, Z)$ independent?

I know that if two random variables are independent, say, $X_1$ and $X_2$, then $h(X_1)$ and $g(X_2)$ are independent, but what about if we have more than two random variables as in the above question?

• I believe you mean $g(Y,Z)$ is a function of only $Y$ and $Z$, in which case yes, they are independent. (Short answer to your question: yes) – parsiad Nov 7 '17 at 5:48
• For the sake of precision you have to say what kind of functions h and g are. If they are Borel measurable functions the answer is YES. – Kavi Rama Murthy Nov 7 '17 at 7:47