Are Functions of Independent Random Variables Always Independent?

[kingwinner]

Homework Statement

Suppose the random variable X has a N(5,25) dsitribution and Y has a N(2,16) distribution and that X and Y are independent. Find a random variable F that is a function of both X and Y such that F has a F-distribution with parameters (1,2), i.e. F(1,2).

Homework Equations

Definition: If X~chi square(n), Y~chi square(m), and X and Y are independent, then (X/n)/(Y/m)~F(n,m)

The Attempt at a Solution

Does F=[(X-5)/5]^2 / {([(X-5)/5]^2 + [(Y-2)/4]^2])/2} work?
The only trouble I am seeing is that (X-5)/5]^2 and [(X-5)/5]^2 + (Y-2)/4]^2] might not be independent. So are they independent? If so, how can I prove it? If not, what else can I do?


Any stat guy here?
I appreciate for any help!

 

[kingwinner]

In other words, we know that if X and Y are independent, then g(X) and h(Y) are independent, but are a function of X (f₁(X)) and a function of X and Y (f₂(X,Y)always independent?