> [!NOTE] Theorem > Let $u\in \mathbb{R}$ and $\sigma>0.$ Let $X$ be [[Continuous random variables|continuous real-valued random variable]] that has [[Normal Distribution|normally distribution]] with parameters $\mu$ and $\sigma^{2}.$ Let $Z=\frac{X-\mu}{\sigma}.$ Then $Z$ has a [[Standard Normal Distribution|standard normal distribution]]. **Proof**: We have $\begin{aligned} \mathbb{P}(a\leqslant Z\leqslant b)& =\mathbb{P}(\mu+a\sigma\leqslant X\leqslant\mu+b\sigma) \\ &=\int_{\mu+a\sigma}^{\mu+a\sigma}\frac1{\sqrt{2\pi\sigma^2}}\cdot e^{-\frac{(x-\mu)^2}{2\sigma^2}}\mathrm{d}x \\ &=\int_a^b\frac1{\sqrt{2\pi}}\cdot e^{-\frac{z^2}2}\mathrm{d}z. \end{aligned}$ **Proof**: ...