Voigt profile

(Centered) Voigt
Probability density function Plot of the centered Voigt profile for four cases. Each case has a full width at half-maximum of very nearly 3.6. The black and red profiles are the limiting cases of the Gaussian (γ =0) and the Lorentzian (σ =0) profiles respectively.
Cumulative distribution function
Parameters	$\gamma ,\sigma >0$
Support	$x\in (-\infty ,\infty )$
PDF	${\frac {\Re [w(z)]}{\sigma {\sqrt {2\pi }}}},~~~z={\frac {x+i\gamma }{\sigma {\sqrt {2}}}}$
CDF	(complicated - see text)
Mean	(not defined)
Median	$0$
Mode	$0$
Variance	(not defined)
Skewness	(not defined)
Excess kurtosis	(not defined)
MGF	(not defined)
CF	$e^{-\gamma \|t\|-\sigma ^{2}t^{2}/2}$

The Voigt profile (named after Woldemar Voigt) is a probability distribution given by a convolution of a Cauchy-Lorentz distribution and a Gaussian distribution. It is often used in analyzing data from spectroscopy or diffraction.