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An exponentially modified Gaussian distribution is often encountered in modelling chromatographic peaks.[ 1] [ 2]
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{\displaystyle f(x)=y_{0}+{\frac {A}{t_{0}}}exp\left({\frac {1}{2}}\left({\frac {w}{t_{0}}}\right)^{2}-{\frac {x-x_{c}}{t_{0}}}\right)\int _{-\infty }^{z}{\frac {1}{\sqrt {2}}}exp\left({\frac {-y^{2}}{z}}\right)dy}
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{\displaystyle f(x)=y_{0}+{\frac {A}{t_{0}}}exp\left({\frac {1}{2}}\left({\frac {w}{t_{0}}}\right)^{2}-{\frac {x-x_{c}}{t_{0}}}\right)\left({\frac {1}{2}}+{\frac {1}{2}}erf\left({\frac {z}{\sqrt {2}}}\right)\right)}
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{\displaystyle y_{0}}
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{\displaystyle A}
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{\displaystyle t_{0}>0}
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{\displaystyle z={\frac {x-x_{c}}{w}}-{\frac {w}{t_{0}}}}
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{\displaystyle erf\left({\frac {z}{\sqrt {2}}}\right)}
error function
The area under the curve (cumulative distribution) is given by[ 3]
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{\displaystyle F(x)={\frac {A}{2t_{0}}}\exp \left({\frac {w^{2}}{2t_{0}^{2}}}+{\frac {x_{c}-x}{t_{0}}}\right)\left[\mathrm {erf} \left({\frac {x-x_{c}}{{\sqrt {2}}w}}-{\frac {w}{{\sqrt {2}}t_{0}}}\right)+1\right]+c_{0}}
which is the same as...
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{\displaystyle F(x)=c_{0}+{\frac {A}{t_{0}}}exp\left({\frac {1}{2}}\left({\frac {w}{t_{0}}}\right)^{2}-{\frac {x-x_{c}}{t_{0}}}\right)\left({\frac {1}{2}}+{\frac {1}{2}}erf\left({\frac {z}{\sqrt {2}}}\right)\right)}
References
Comment: As this submission had no inline references, help was added to the author's talk page, User talk:Quantumkinetics ChzzBot IV (talk) 15:15, 29 November 2011 (UTC)
![{\displaystyle F(x)={\frac {A}{2t_{0}}}\exp \left({\frac {w^{2}}{2t_{0}^{2}}}+{\frac {x_{c}-x}{t_{0}}}\right)\left[\mathrm {erf} \left({\frac {x-x_{c}}{{\sqrt {2}}w}}-{\frac {w}{{\sqrt {2}}t_{0}}}\right)+1\right]+c_{0}}](/media/api/rest_v1/media/math/render/svg/568105fc247306ab2f6ef153ba594e0a8be1676b)
