Let $X$ be a normal random variable with parameters $\mu\in\mathbb R$ and $\sigma^2>0$. Find the cdf and pdf for $Z$, the standardization of $X$.
What approach should I take on this? I initially thought of using the pdf of a normal distribution, but again, the question requires I find the pdf.
The standartization of $X$ is given by $$ Z=\frac{X-\mu}{\sigma}, $$ where $\mu=\operatorname EX$ and $\sigma^2=\operatorname{Var}X$. We have that $$ P(Z\le z)=P\biggl(\frac{X-\mu}{\sigma}\le z\biggr)=P(X\le \sigma z+\mu) $$ and we know that $X\sim\mathcal N(\mu,\sigma^2)$. Hence, $$ P(X\le \sigma z+\mu)=\frac1{\sigma\sqrt{2\pi}}\int_{-\infty}^{\sigma z+\mu}\exp\Bigl\{ -\frac{(x-\mu)^2}{2\sigma^2}\Bigr\}\mathrm dx. $$ Using the substitution $x=\sigma t+\mu$, $$ P(X\le \sigma z+\mu)=\frac1{\sqrt{2\pi}}\int_{-\infty}^z\exp\Bigl\{ -\frac{t^2}2\Bigr\}\mathrm dt. $$ This shows that $Z\sim\mathcal N(0,1)$, i.e. $Z$ has the standard normal distribution.