3.6: The Gaussian PDF

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Page ID: 47237

Franz S. Hover & Michael S. Triantafyllou
Massachusetts Institute of Technology via MIT OpenCourseWare

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The normal or Gaussian pdf is one of the most popular distributions for describing random variables, partly because many physical systems do exhibit Gaussian variability, and partly because the Gaussian pdf is amenable to some very powerful tools in design and analysis. It is

\[ \underline{p}(x) = \dfrac{1}{\sigma \sqrt{2 \pi}} e^{(x - \bar{x})^2 \ 2 \sigma^2}, \]

where \( \sigma\) and \( \sigma^2\) are the standard deviation and variance, respectively, and \( \bar{x}\) is the mean value. By design, this pdf always has an area that equals one. The cumulative probability function is

\[P(x) = \dfrac{1}{2} + \textrm{erf} \left( \dfrac{x - \bar{x}}{\sigma} \right)\]

where

\[\textrm{erf} (\xi) = \dfrac{1}{\sqrt {2 \pi}} \int\limits_{0}^{\xi} e ^{-\xi^2 /2} \, d\xi. \]

Don’t try to compute the error function erf(); look it up in a table or call a subroutine! The Gaussian distribution has a shorthand: \( N(\bar{x} , \sigma^2) \). The arguments are the mean and variance.