Normal Distributions

Caution

This page is in notes format, and may not be of the same quality as other pages on this site.

Probability Density Function

The probability density function of a normal distribution is given by:

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

where:
$\mu$ is the mean of the distribution
$\sigma$ is the standard deviation

A few example probability density functions (PDFs) for the normal distribution.

See https://github.com/gbmhunter/BlogAssets/tree/master/Mathematics/Statistics/NormalDistribution for the code which generated these graphs.

Capped Normal Distribution

$$\sigma_{overall}^2 = \frac{1}{\sqrt{2\pi} \sigma_{nd} + c} \left [\int_{-\infty}^{-c} +\int_{c}^{\infty} x^2 e^{(-\frac{1}{2} (\frac{x - c}{\sigma_{nd}})^2)} dx + \int_{-c}^{c} x^2 dx \right ]$$

where:
$\sigma_{overall}$ is the standard deviation of the capped distribution
$\sigma_{nd}$ is the standard deviation of the normal distribution tails, ignoring the flat section in the middle
$c$ is the half-width of the flat section of the capped distribution
$x$ is the random variable