Showing independence of random variables When proving $\bar x$ and $S^2$ are independent in my noted it says that "functions of independent quantities are independent ". Can someone tell me how functions of independent quantities are independent happen?
Also let $X_1,X_2,X_3,\dots,X_n$ be a random sample.And suppose we want to estimate parameter $\theta$ using $T(x)$ as an estimator.
In this I have a expression as $E\{ [  T(X)-E(T(X))][E(T(x)-\theta)]\}$.
I want to know if I can say that in $[T(X)-E(T(X))]$ since $E(T(X)$ (expected value is a one particular fixed value) and $T(X)$ are independent of one another.
Also the expression $[E(T(x)-\theta)]$ is independent of $X$.
Hence the two expressions $[T(X)-E(T(X))]$ and $[E(T(x)-\theta)]$  are independent of one another. So that I can use if $a,b$ are independent $E(ab)=E(a)E(b)$ form

For your first question, suppose $X$ and $Y$ are independent random variables.
The statement is that for any Borel measurable functions $f$ and $g$, $f(X)$ and $g(Y)$ are independent.  In fact, independence of $X$ and $Y$ is equivalent to
(and in some formulations is defined as) the events $X \in A$ and $Y \in B$ being independent for all Borel subsets $A$ and $B$ of $\mathbb R$.  The result is then immediate, since $f(X) \in A$ iff $X$ is in the Borel set $f^{-1}(A)$, and similarly for $g(Y) \in B$. 
