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Error Function distribution

From phenomenon to model

The error function erf arises when integrating the Gaussian bell and appears in diffusion, heat equations, and error theory. Distribution catalogues use this name for a normal form parameterized through that special function.

What to look at

It spans the real line and has mean zero. After an appropriate rescaling it has only one shape degree of freedom; its name comes from the special function in its CDF.

Reading Result
support entire real line
centre zero
larger h greater concentration
family reparameterized normal

Names you may also encounter

It may be called erf distribution or a normal distribution parameterized through h. Do not confuse it with a measurement-error distribution in general or with the complementary error function.

When it makes sense

It can be useful pedagogically when a derivation produces erf directly, or when a bell curve is fit using h instead of standard deviation. For communication, translating to a familiar normal scale is helpful.

What not to assume

When comparing h with sigma, do not equate the numbers without checking the exact density convention used by the implementation.

A Normal law written with its namesake integral

The error function emerged from integrating the bell e^-x² and now connects diffusion, heat conduction, and Gaussian probability. In Phitter, Error Function does not introduce a new shape: it is a centred Normal whose standard deviation is the inverse of the product of h and the square root of two.

The parameter h is an inverse scale, so larger values narrow the distribution. This convention appears in physics texts that work directly with erf. Recognizing the equivalence prevents treating Error Function and Normal as rival families or counting the same fit twice. Its value lies in translating notation across disciplines, especially when solutions of the heat equation are read as cumulative probabilities.

Decision guide

A good candidate when: the derivation or application naturally expresses a centred Normal law through erf and the inverse-scale parameter h.

Compare it with: Normal. They represent the same shape family in this implementation; convert h to standard deviation before comparing results or communicating uncertainty.

References

Error Function Distribution: equations and calculator

Distribution defintion

X∼ErrorFunction(h)X\sim\mathrm{ErrorFunction}\left(h\right)

Distribution domain

x∈(−∞,∞)x\in\left(-\infty,\infty\right)

Parameters domain and parameters constraints

h∈R+h\in\mathbb{R}^{+}

Cumulative distribution function

FX(x)=Φ(2hx)F_{X}\left(x\right)=\Phi( \sqrt{2}hx )

Probability density function

fX(x)=hπe−h2x2f_{X}\left(x\right)=\frac{h}{\sqrt{\pi}}e^{-h^{2}x^{2}}

Percent point function/Sample

FX−1(u)=Φ−1(u)2hF^{-1}_{X}\left(u\right)=\frac{\Phi^{-1}(u)}{\sqrt{2}h}

Non-central parametric moments

μk′=E[Xk]=∫−∞∞xkfX(x)dx\mu'_{k}=E[X^k]=\int_{-\infty }^{\infty }x^{k}f_{X}\left(x\right)dx

Parametric mean

Mean(X)=μ1′=0\mathrm{Mean}(X)=\mu'_{1}=0

Parametric variance

Variance(X)=μ2′−μ1′2=12h2\mathrm{Variance}(X)=\mu'_{2}-\mu'^{2}_{1}=\frac{1}{2h^{2}}

Parametric skewness

Skewness(X)=μ3′−3μ2′μ1′+2μ1′3(μ2′−μ1′2)1.5=0\mathrm{Skewness}(X)=\frac{\mu'_{3}-3\mu'_{2}\mu'_{1}+2\mu'^{3}_{1}}{(\mu'_{2}-\mu'^{2}_{1})^{1.5}}=0

Parametric kurtosis

Kurtosis(X)=μ4′−4μ1′μ3′+6μ1′2μ2′−3μ1′4(μ2′−μ1′2)2=3\mathrm{Kurtosis}(X)=\frac{\mu'_{4}-4\mu'_{1}\mu'_{3}+6\mu'^{2}_{1}\mu'_{2}-3\mu'^{4}_{1}}{(\mu'_{2}-\mu'^{2}_{1})^{2}}=3

Parametric median

Median(X)=0\mathrm{Median}(X)=0

Parametric mode

Mode(X)=0\mathrm{Mode}(X)=0

Additional information and definitions

h:Inverse of scale parameterh:\text{Inverse of scale parameter}
u:Uniform[0,1] random varibleu:\text{Uniform[0,1] random varible}
Φ(x):CDF normal standard distribution\Phi\left(x\right):\text{CDF normal standard distribution}