Standardizing *any* random variable. If $X$ is any random variable, and if $Z=\dfrac{X-\mu}{\sigma}$, then are the following true or false:


*

*The mean of $Z$ is always $0$, regardless of the distribution of $X$.

*The variance of $Z$ is always $1$, regardless of the distribution of $X$.

*$Z$ is always normally distributed, regardless of the distribution of $X$.


I ran some quick calculations, and I'am persuaded that the first two statements are true. For statement #3, my intuition says the answer is false, but I don't really know how to formulate an answer.
Is this a central limit theorem type question?
 A: For (1):
Subtracting the mean from X will result in E(x-E(x)) = E(x) - E(x) = 0
For (2):
Proof by counter-example (similar to the answer above)
For (3):
Proof by counter-example (as comment above), take X as the Uniform distribution for example.
Multiplying X by 1/(sigma) and subtracting (mu)/(sigma) does not make X normal.
A: $\newcommand{\var}{\operatorname{Var}}$You first need $E[X]$ and $E[X^2]$ to exist (i.e. $E[X^2] < \infty$).

*

*Use linearity of expectation to show its true.

*Use linearity of expectation to show $\var(Z) = E[(Z-0)^2] = E[Z^2] = 1$.

*Obviously not. Pretty much any non-Gaussian distribution will have this not be Gaussian (e.g. a Bernoulli rv will become $\frac{1}{\sqrt{2}}$ times a rademacher).

A: A simple counterexample suffices for claim #3:  just let $X \sim \mathrm{Bernoulli}(0.5)$.  Since $X$ is discrete, $Z = (X - \mu)/\sigma$ is also discrete, which contradicts the claim that $Z$ is normal (hence, continuous).
Of course, even if we restrict the claim to continuous random variables with infinite support, another counterexample is the exponential distribution:  $f_X(x) = e^{-x}$, for $x > 0$.  And if $X$ is required to be continuous with support over the entire real line, then choose the double exponential distribution $$f_X(x) = \frac{1}{2} e^{-|x|}.$$  The cusp at $f(0)$ cannot be removed by standardization, or indeed, by any nondegenerate location-scale transformation.
