Exponentially Modified Gaussian

An exponentially modified Gaussian distribution is often encountered in modelling chromatographic peaks, in modelling cell proliferation and differentiation, and in modelling cluster ion beams. The form of the probability distribution equation is given below.^, Other forms of the equation have been published using slightly different notation or replacing the error function term with a hyperbolic tangent term, which is an approximation.

$f(x)=y_{0}+{\frac {A}{t_{0}}}exp\left({\frac {1}{2}}\left({\frac {w}{t_{0}}}\right)^{2}-{\frac {x-x_{c}}{t_{0}}}\right)\int _{-\infty }^{z}{\frac {1}{\sqrt {2}}}exp\left({\frac {-y^{2}}{z}}\right)dy$

or

$f(x)=y_{0}+{\frac {A}{t_{0}}}exp\left({\frac {1}{2}}\left({\frac {w}{t_{0}}}\right)^{2}-{\frac {x-x_{c}}{t_{0}}}\right)\left({\frac {1}{2}}+{\frac {1}{2}}erf\left({\frac {z}{\sqrt {2}}}\right)\right)$

$y_{0}$ = the initial value

$A$ = the amplitude

$x_{c}$ = the center of the peak

$w$ = the width of the peak

$t_{0}$ = the modification factor (skewness, $t_{0}>0$)

$z={\frac {x-x_{c}}{w}}-{\frac {w}{t_{0}}}$

$erf\left({\frac {z}{\sqrt {2}}}\right)$ = the error function of ${\frac {z}{\sqrt {2}}}$