Exponentially modified Gaussian distribution

EMG
EMG
	Probability density function
	Cumulative distribution function
Parameters	μ ∈ R — mean of Gaussian component; σ2 > 0 — variance of Gaussian component; λ > 0 — rate of exponential component
Support	x ∈ R
PDF
CDF	; where; is the CDF of a Gaussian distribution
Mean
Mode
Variance
Skewness
Excess kurtosis
MGF
CF

In probability theory, an exponentially modified Gaussian distribution (EMG, also known as exGaussian distribution) describes the sum of independent normal and exponential random variables. An exGaussian random variable Z may be expressed as Z = X + Y, where X and Y are independent, X is Gaussian with mean μ and variance σ², and Y is exponential of rate λ. It has a characteristic positive skew from the exponential component.

It may also be regarded as a weighted function of a shifted exponential with the weight being a function of the normal distribution.

EMG
Probability density function
Cumulative distribution function
Parameters	μ ∈ R — mean of Gaussian component σ² > 0 — variance of Gaussian component λ > 0 — rate of exponential component
Support	x ∈ R
PDF	${\frac {\lambda }{2}}\exp \left[{\frac {\lambda }{2}}(2\mu +\lambda \sigma ^{2}-2x)\right]\operatorname {erfc} \left({\frac {\mu +\lambda \sigma ^{2}-x}{{\sqrt {2}}\sigma }}\right)$
CDF	$\Phi (x,\mu ,\sigma )-{\frac {1}{2}}\exp \left[{\frac {\lambda }{2}}(2\mu +\lambda \sigma ^{2}-2x)\right]\operatorname {erfc} \left({\frac {\mu +\lambda \sigma ^{2}-x}{{\sqrt {2}}\sigma }}\right)$ where $\Phi (x,\mu ,\sigma )$ is the CDF of a Gaussian distribution
Mean	$\mu +1/\lambda$
Mode	$x_{m}=\mu -\operatorname {sgn} \left(\tau \right){\sqrt {2}}\sigma \operatorname {erfcxinv} \left({\frac {{\|}\tau {\|}}{\sigma }}{\sqrt {\frac {2}{\pi }}}\right)+{\frac {\sigma ^{2}}{\tau }}$ $f(x_{m})=h\exp \left(-{\frac {1}{2}}\left({\frac {\mu -x_{m}}{\sigma }}\right)^{2}\right)$
Variance	$\sigma ^{2}+1/\lambda ^{2}$
Skewness	${\frac {2}{\sigma ^{3}\lambda ^{3}}}\left(1+{\frac {1}{\sigma ^{2}\lambda ^{2}}}\right)^{-3/2}$
Excess kurtosis	${\frac {3\left(1+{\frac {2}{\sigma ^{2}\lambda ^{2}}}+{\frac {3}{\lambda ^{4}\sigma ^{4}}}\right)}{\left(1+{\frac {1}{\lambda ^{2}\sigma ^{2}}}\right)^{2}}}-3$
MGF	$\left(1-{\frac {t}{\lambda }}\right)^{-1}\,\exp \left(\mu t+{\frac {1}{2}}\sigma ^{2}t^{2}\right)$
CF	$\left(1-{\frac {it}{\lambda }}\right)^{-1}\,\exp \left(i\mu t-{\frac {1}{2}}\sigma ^{2}t^{2}\right)$