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Normal Distributions

Published On:
Aug 30, 2018
Last Updated:
Aug 30, 2018

Probability Density Function

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

f(x,μσ2)=12πσ2e(xμ)22σ2f(x, \mu | \sigma^2) = \frac{1}{\sqrt{2\pi \sigma^2}} e^{-\frac{(x - \mu)^2}{2\sigma^2}}

where:
μ\mu is the mean of the distribution
σ\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

σoverall2=12πσnd+c[c+cx2e(12(xcσnd)2)dx+ccx2dx]\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:
σoverall\sigma_{overall} is the standard deviation of the capped distribution
σnd\sigma_{nd} is the standard deviation of the normal distribution tails, ignoring the flat section in the middle
cc is the half-width of the flat section of the capped distribution
xx is the random variable