Are the absolute values of random variables iid if the random variables are iid? If $X$ and $Y$ are independent and identically distributed (iid) random variables, does it imply that $|X|$ and $|Y|$ are iid? How would you go about proving this?
 A: $X,Y$ are independent if and only if $$P\left\{ X\in A\wedge Y\in B\right\} =P\left\{ X\in A\right\} P\left\{ Y\in B\right\} $$
is true for measurable sets $A,B$.
If this is the case and $f,g:\mathbb{R}\rightarrow\mathbb{R}$ are
measurable functions then:
$$P\left\{ f\left(X\right)\in A\wedge g\left(Y\right)\in B\right\} =P\left\{ X\in f^{-1}\left(A\right)\wedge Y\in g^{-1}\left(B\right)\right\} =P\left\{ X\in f^{-1}\left(A\right)\right\} P\left\{ Y\in g^{-1}\left(B\right)\right\} =P\left\{ f\left(X\right)\in A\right\} P\left\{ g\left(Y\right)\in B\right\} $$
showing that $f\left(X\right)$ and $g\left(Y\right)$ are independent.
A: Yes. In fact one can show that $X$ and $Y$ are independent if and only if $f(X)$ and $g(Y)$ are independent for every measurable  function $f, g$ 
I don't know how you introduced independence, but X and Y independent means that the $\sigma$-algebrae generated by them are independent; it is easy to see that the $\sigma$-algebra generated by $f(X)$ is smaller that the one generated by $X$, hence independence is preserved 
A: Evidently so. See this link 
Are functions of independent variables also independent?
which addresses independence. It's clear they are identically distributed, isn't it?
