Continuous Distributions

1 min read Last updated Sat Jun 27 2026 08:46:52 GMT+0000 (Coordinated Universal Time)

Normal Distribution

Denoted by $N(\mu, \sigma^2)$ .

f(x) = \frac{1}{\sqrt{2\pi\sigma^2}} \exp\left(-\frac{(x-\mu)^2}{2\sigma^2}\right)

Properties:

Symmetric about $\mu$
Kurtosis is 3
CDF has a sigmoid (S-shaped) curve; no closed form; use tables or software

When to Use

Applies when the Central Limit Theorem (CLT) holds: sample means from any distribution with finite variance approach normality as $n \to \infty$ . Rule of thumb: $n \geq 30$ .

Standard Normal Distribution

Denoted by $N(0, 1)$ . Variable is written as $z$ .

\phi(z) = \frac{1}{\sqrt{2\pi}} \exp\left(-\frac{z^2}{2}\right)

Standardization

Any $X \sim N(\mu, \sigma^2)$ can be converted to $z$ :

z = \frac{x - \mu}{\sigma}

Standardization enables use of a single $z$ -table for all normal distributions.

Other continuous distributions are explained in separate pages, in detail.

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