[Math] how to prove standardizing a normally distributed random variable

probabilityrandom variables

  1. 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.

  2. 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?

Best Answer

There is no need to use PDFs.

  1. 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*}

  2. 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.

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