Gaussian Random Variables

calendar_today March 31, 2026 article 126 words access_time 2 min replay Modified March 31, 2026

Table of Contents

Gaussian $(\mu,\sigma)$

Definition 19.1 (Gaussian Random Variable).

$X$ is a Gaussian $(\mu,\sigma)$ random variable if the PDF of $X$ is

$$ f_X(x)=\frac{1}{\sqrt{2\pi}\sigma}\,e^{-\frac{(x-\mu)^2}{2\sigma^2}} $$

where $\mu\in\mathbb R$ and $\sigma>0$

Theorem 19.2 (Gaussian Mean Variance).

If $X$ is Gaussian $(\mu,\sigma)$

$$ E[X]=\mu $$$$ \mathrm{Var}[X]=\sigma^2 $$

Theorem 19.3 (Affine Transform of Gaussian).

If $X$ is Gaussian $(\mu,\sigma)$ and $Y=aX+b$ then $Y$ is Gaussian $(a\mu+b,|a|\sigma)$

Definition 19.4 (Standard Normal Random Variable).

The standard normal random variable $Z$ is the Gaussian $(0,1)$ random variable

Definition 19.5 (Standard Normal CDF).

The CDF of $Z$ is

$$ \Phi(z)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{z} e^{-u^2/2}\,du $$

Theorem 19.6 (Standardization).

If $X$ is Gaussian $(\mu,\sigma)$ then

$$ Z=\frac{X-\mu}{\sigma} $$

is standard normal

Theorem 19.7 (Gaussian CDF via Standard Normal).

If $X$ is Gaussian $(\mu,\sigma)$ then

$$ F_X(x)=\Phi\left(\frac{x-\mu}{\sigma}\right) $$

and for $a $$ P[a

Theorem 19.8 (Standard Normal Symmetry).

$$ \Phi(-z)=1-\Phi(z) $$

Corollary 19.9 (Symmetric Interval Probability).

For $z>0$

$$ P[-z

For a Gaussian random variable
- about $68\%$ of outcomes are within $1\sigma$ of $\mu$
- about $95\%$ of outcomes are within $2\sigma$ of $\mu$
- about $99.7\%$ of outcomes are within $3\sigma$ of $\mu$

Definition 19.10 (Complementary CDF).

The standard normal complementary CDF is

$$ Q(z)=P[Z>z]=\frac{1}{\sqrt{2\pi}}\int_{z}^{\infty} e^{-u^2/2}\,du=1-\Phi(z) $$

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