Editing Error function (section)

==Applications==
When the results of a series of measurements are described by a [[normal distribution]] with [[standard deviation]] {{mvar|σ}} and [[expected value]] 0, then {{math|erf ({{sfrac|''a''|''σ'' {{sqrt|2}}}})}} is the probability that the error of a single measurement lies between {{math|−''a''}} and {{math|+''a''}}, for positive {{mvar|a}}. This is useful, for example, in determining the [[bit error rate]] of a digital communication system.

The error and complementary error functions occur, for example, in solutions of the [[heat equation]] when [[boundary condition]]s are given by the [[Heaviside step function]].

The error function and its approximations can be used to estimate results that hold [[with high probability]] or with low probability. Given a random variable {{math|''X'' ~ Norm[''μ'',''σ'']}} (a normal distribution with mean {{mvar|μ}} and standard deviation {{mvar|σ}}) and a constant {{math|''L'' > ''μ''}}, it can be shown via integration by substitution:
<math display="block">\begin{align}
\Pr[X\leq L] &= \frac{1}{2} + \frac{1}{2} \operatorname{erf}\frac{L-\mu}{\sqrt{2}\sigma} \\
&\approx A \exp \left(-B \left(\frac{L-\mu}{\sigma}\right)^2\right)
\end{align}</math>

where {{mvar|A}} and {{mvar|B}} are certain numeric constants. If {{mvar|L}} is sufficiently far from the mean, specifically {{math|''μ'' − ''L'' ≥ ''σ''{{sqrt|ln ''k''}}}}, then:

<math display="block">\Pr[X\leq L] \leq A \exp (-B \ln{k}) = \frac{A}{k^B}</math>

so the probability goes to 0 as {{math|''k'' → ∞}}.

The probability for {{mvar|X}} being in the interval {{closed-closed|''L<sub>a</sub>'', ''L<sub>b</sub>''}} can be derived as
<math display="block">\begin{align}
\Pr[L_a\leq X \leq L_b] &= \int_{L_a}^{L_b} \frac{1}{\sqrt{2\pi}\sigma} \exp\left(-\frac{(x-\mu)^2}{2\sigma^2}\right) \,\mathrm dx \\
&= \frac{1}{2}\left(\operatorname{erf}\frac{L_b-\mu}{\sqrt{2}\sigma} - \operatorname{erf}\frac{L_a-\mu}{\sqrt{2}\sigma}\right).\end{align}</math>