MIT 18.650 — Lecture 4

The Normal Distribution

Definition. The normal distribution $X \sim \mathcal{N}(\mu, \sigma^2)$ has PDF given by $f(x) = \dfrac{1}{\sigma \sqrt{2\pi}} \exp\left(-\frac{(x - \mu)^2}{2 \sigma^2}\right)$ .

Definition. For any $X \sim \mathcal{N}(\mu, \sigma^2)$ , we say $Z = \frac{X - \mu}{\sigma}$ is the z-score of $X$ .

Note that if $X \sim \mathcal{N}(\mu, \sigma^2)$ , then $aX + b \sim \mathcal{N}(a \mu + b, a^2 \sigma^2)$ , expectedly. Therefore,

Theorem. (Z-Score) The random variable $Z \sim \mathcal{N}(0, 1)$ has PDF given by $f(x) = \frac{1}{\sqrt{2\pi}}e^{-x^2/2}$ and CDF $\Phi(x) := \int_{-\infty}^x f(t) \, \mathrm{d}t$ .

Example. Given that $X \sim \mathcal{N}(4, 4)$ , compute $\Pr(0 \leq X \leq 10)$ .

Solution: Considering the z-score $Z = \frac{X - 4}{2}$ , this is just $\Pr(-2 \leq Z \leq 3)$ , or $\Phi(3) - \Phi(-2)$ .