Showing independence of random variables

Question

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

Answer 1

I'm not sure I understand your questions, but let's see if this can answer part of it.

$$ \begin{bmatrix} X_1 \\ \vdots \\ X_n \end{bmatrix} = \begin{bmatrix} \bar X \\ \vdots \\ \bar X \end{bmatrix} + \begin{bmatrix} X_1-\bar X \\ \vdots \\ X_n-\bar X \end{bmatrix} $$

Suppose $X_1,\ldots,X_n \sim \mathrm{i.i.d.}\ N(\mu,\sigma^2)$, Then $\bar X$ is uncorrelated with $X_i-\bar X$:

$$ \operatorname{cov}(\bar X, X_i-\bar X) = \operatorname{cov}(\bar X, X_i) - \operatorname{var}(\bar X) = \frac{\sigma^2}{n} - \frac{\sigma^2}{n} = 0. $$

If two random vectors are jointly normal and uncorrelated, then they are independent. Thus $\bar X$ and the last vector on the right side of the equality above are independent. And $S^2$ is a function of that vector, so $\bar X$ and $S^2$ are independent.

Without the assumption that $X_1,\ldots,X_n \sim \mathrm{i.i.d.}\ N(\mu,\sigma^2)$, you won't get all of this.

Answer 2

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 $g(Y) \in B$.