# Mutual independence of functions of random variables

In a related question, it is noted that two functions of independent random variables are themselves independent.

Does this result extend to three or more functions of independent random variables? Also, are the functions necessarily mutually independent?

Consider the following pretty well known example. Assume $X_1,X_2 \sim \text{Gamma}(\alpha, \beta)$. That means the original density is $$f_{X_1,X_2}(x_1,x_2) = \frac{1}{\Gamma(\alpha)^2 \beta^{2\alpha}}x_1^{\alpha-1}x_2^{\alpha-1}\exp\left[-\frac{x_1+x_2}{\beta}\right].$$ Then define $Y_1 = X_1/(X_1 + X_2)$ and $Y_2 = X_1 + X_2$. Then the new joint density is $$f_{Y_1,Y_2}(y_1,y_2) = \left[\frac{\Gamma(2\alpha)}{\Gamma(\alpha)^2 }y_1^{\alpha-1}(1-y_1)^{\alpha-1} \right]\left[\frac{1}{\beta^{2\alpha}\Gamma(2\alpha)}y_2^{2\alpha-1}\exp\left[-\frac{y_2}{\beta}\right] \right].$$ $Y_1 \sim \text{Beta}(\alpha,\alpha)$, $Y_2 \sim \text{Gamma}(2\alpha,\beta)$, and they are independent.
• So in other words, we can avoid thinking about mutual independence for $n$ random variables by making a function of $n-1$ random variables and realizing that it must be independent of a function of the other random variable (provided that the $n$ random variables are mutually independent to begin with)? – Taliant Mar 5 '18 at 5:59