how do I prove standardizing a normally distributed random variable 
*

*Let X be a random variable of mean $\mu$ and variance $\sigma^2$. Use the properties of expectation to show that
$$Z=\frac{X-\mu}{\sigma}$$
has mean 0 and variance 1. 

*Let Z be a random variable of mean 0 and variance 1. Show that 
$$X=\sigma Z+\mu$$
has mean $\mu$ and variance $\sigma^2$.  
I think I have to use $f_Z(z)=\frac{1}{\sqrt{2\pi}}e^{\frac{-z^2}{2}}$ to help me prove this but I have no idea where to start? I know what expectation is and how to calculate it but which properties specifically is the question talking about?
 A: There is no need to use PDFs.


*

*Since you're only allowed to use properties of expectation, then
$$E[Z] = E\left[\frac{X-\mu}{\sigma}\right] = \frac{1}{\sigma}\left[E[X] -   
\mu\right] = \frac{1}{\sigma}[\mu-\mu] = 0.$$
Then for the variance
\begin{align*}
\text{Var}[Z^2] &= E[Z^2]-E^2[Z] \\
&= E\left[\left(\frac{X-\mu}{\sigma}\right)^2\right] - 0^2\\
&= \frac{1}{\sigma^2}\left[E[X^2]-2\mu E[X] +\mu^2\right] \\
&= \frac{1}{\sigma^2}[\{E[X^2]-E^2[X]\} + E^2[X]-2\mu E[X]+\mu^2]\\
&= \frac{1}{\sigma^2}[\{\text{Var}[X]\}+\mu^2-2\mu^2+\mu^2]\\
&= \frac{1}{\sigma^2}[\sigma^2 +0] \\
&= 1.
\end{align*} 

*Similar as 1.

Addendum: If $X,Y$ (not necessarily independent) are random variables and $a,b, c$ are some constants, then the properties it is referring to are


*

*Scaling:
$$E[cX] =cE[X].$$

*Addition: 
$$E[X+Y] = E[X]+E[Y].$$
Then $$E[aX+bY+c] = aE[X]+bE[Y]+ c.$$
The instructions imply that you should be familiar with this by now.
A: For question 1, you cans use $E[\frac{X-\mu}{\sigma}]=E[\frac{X}{\sigma}]-\frac{\mu}{\sigma}=\frac{\mu}{\sigma}-\frac{\mu}{\sigma}=0$.
For the variance, use the variance properties: $Var(\frac{X-\mu}{\sigma})=Var(\frac{X}{\sigma}-\frac{\mu}{\sigma})=Var(\frac{X}{\sigma})=\frac{1}{\sigma^2}Var(X)=\frac{\sigma^2}{\sigma^2}=1$.
Question 2 can be answered in a similar manner.
