Normal Distribution

Compact study note.

Summary

The normal distribution is a symmetric continuous distribution central to measurement error, approximation theory, and the central limit theorem.^[1]

Prerequisites

• Continuous Random Variable

Definition

$X\sim N(\mu ,{\sigma }^2)$ with mean parameter $\mu$ and variance parameter ${\sigma }^2$ .

Notation and Assumptions

$\sigma >0$ . The standard normal is $Z\sim N(0,1)$ .

Parameters

$\mu \in \mathbb{R}$ and $\sigma >0$ .

Support

$\mathbb{R}$ .

PMF or PDF

$f_X(x)=\frac{1}{\sigma \sqrt{2\pi }}\exp [-(x-\mu {)}^2/(2{\sigma }^2)]$ .

CDF

$F_X(x)=\mathrm{\Phi }((x-\mu )/\sigma )$ , where $\mathrm{\Phi }$ is the standard normal CDF.

Moments

$E[X]=\mu$ , $\mathrm{Var}(X)={\sigma }^2$ , and $M_X(t)=\exp (\mu t+{\sigma }^2{t}^2/2)$ .

Essential Result

Standardization converts $X$ to $Z=(X-\mu )/\sigma \sim N(0,1)$ .

Small Example

If $X\sim N(70,{15}^2)$ , then $P(X\le 85)=\mathrm{\Phi }(1)\approx 0.8413$ .

Common Mistakes

• Writing $N(\mu ,\sigma )$ when the convention is $N(\mu ,{\sigma }^2)$ without saying so.
• Using normal approximations without checking scale and support.

Connections

• Central Limit Theorem
• Log-Normal Distribution
• Chi-Square Distribution

References