exGaussian

The ex-Gaussian distribution is a probability distribution that arises from the sum of an exponential distribution and a Gaussian (normal) distribution. It is often used in modeling reaction times and other cognitive processes where there are both random variability and a systematic offset.

Mathematically, the ex-Gaussian distribution is a convolution of an exponential distribution with probability density function $f_E(x; \lambda) = \lambda e^{-\lambda x}$ for $x \ge 0$ and a Gaussian distribution with probability density function $f_G(x; \mu, \sigma^2) = \frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{1}{2}(\frac{x-\mu}{\sigma})^2}$. The resulting probability density function of the ex-Gaussian distribution is given by:

$f_{exG}(x; \lambda, \mu, \sigma^2) = \lambda e^{\lambda(\mu + \frac{\lambda \sigma^2}{2})} \left[1 - \Phi\left(\frac{x - \mu - \lambda \sigma^2}{\sigma}\right)\right]$

where $\Phi$ is the cumulative distribution function of the standard normal distribution.

The ex-Gaussian distribution has three parameters: $\lambda$ (the rate parameter of the exponential component), $\mu$ (the mean of the Gaussian component), and $\sigma^2$ (the variance of the Gaussian component). The parameter $\lambda$ controls the skewness of the distribution, with larger values leading to a more skewed distribution. The parameters $\mu$ and $\sigma^2$ influence the location and spread of the Gaussian component.

This distribution is particularly useful when data exhibits a mixture of processes, where one component is exponentially distributed (often representing simple, quick responses) and another is normally distributed (representing more complex, variable responses). Its applications are found in fields such as psychology, neuroscience, and psychometrics.