Three Variable Transformation and Independence

Question

If $X_1$, $X_2$, and $X_3$ are independent, identically distributed random variables and $Y_1$, $Y_2$, and $Y_3$ are functions of them, how do I show that the $Y$ variables are mutually independent? Can I show $f(Y_1,Y_2,Y_3)=f(Y_1)f(Y_2)f(Y_3)$, or do I have to do it pairwise, ie. $f(Y_1,Y_2)=f(Y_1)f(Y_2)$ etc.?

Answer 1

That's one way to do it. You don't need to do it pairwise if you want to show all three are independent of one another. In fact pairwise independent does not imply independence.

So all in all, you want to show

$$f_{Y_1,Y_2,Y_3}(y_1,y_2,y_3) = f_{Y_1}(y_1)f_{Y_2}(y_2)f_{Y_3}(y_3)$$