Error function: Difference between revisions
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In [[mathematics]], the '''error function''' is a [[function (mathematics)|function]] associated with the [[cumulative distribution function]] of the [[normal distribution]]. | In [[mathematics]], the '''error function''' is a [[function (mathematics)|function]] associated with the [[cumulative distribution function]] of the [[normal distribution]]. | ||
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:<math>\operatorname{erf}(x) = \frac{2}{\sqrt\pi} \int_{0}^{x} \exp(-t^2) dt .\,</math> | :<math>\operatorname{erf}(x) = \frac{2}{\sqrt\pi} \int_{0}^{x} \exp(-t^2) dt .\,</math> | ||
The '''complementary error function''' is defined as | |||
:<math>\operatorname{erfc}(x) = 1 - \operatorname{erf}(x) .\,</math> | |||
The probability that a normally distributed random variable ''X'' with mean μ and variance σ<sup>2</sup> exceeds ''x'' is | The probability that a normally distributed random variable ''X'' with mean μ and variance σ<sup>2</sup> exceeds ''x'' is | ||
:<math>F(x;\mu,\sigma)=\frac{1}{2} \left[ 1 + \operatorname{erf} \left( \frac{x-\mu}{\sigma\sqrt{2}} \right) \right]. | :<math>F(x;\mu,\sigma)=\frac{1}{2} \left[ 1 + \operatorname{erf} \left( \frac{x-\mu}{\sigma\sqrt{2}} \right) \right]. | ||
</math> | </math>[[Category:Suggestion Bot Tag]] | ||
Latest revision as of 12:00, 13 August 2024
In mathematics, the error function is a function associated with the cumulative distribution function of the normal distribution.
The definition is
The complementary error function is defined as
The probability that a normally distributed random variable X with mean μ and variance σ2 exceeds x is
![{\displaystyle F(x;\mu ,\sigma )={\frac {1}{2}}\left[1+\operatorname {erf} \left({\frac {x-\mu }{\sigma {\sqrt {2}}}}\right)\right].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/df6f63a0a3433d305d9e626f9a8e03c67bd1536a)