The ex-Gaussian distribution is often used by psychologists to model response time (RT). It is defined by adding two
random variables, one from a normal distribution and the other from an exponential. The parameters `mu`

and
`sigma`

are the mean and standard deviation from the normal distribution variable while the parameter `nu`

is the mean of the exponential variable.
The functions `dexGAUS`

, `pexGAUS`

, `qexGAUS`

and `rexGAUS`

define the density, distribution function,
quantile function and random generation for the ex-Gaussian distribution.

```
exGAUS(mu.link = "identity", sigma.link = "log", nu.link = "log")
dexGAUS(x, mu = 5, sigma = 1, nu = 1, log = FALSE)
pexGAUS(q, mu = 5, sigma = 1, nu = 1, lower.tail = TRUE, log.p = FALSE)
qexGAUS(p, mu = 5, sigma = 1, nu = 1, lower.tail = TRUE, log.p = FALSE)
rexGAUS(n, mu = 5, sigma = 1, nu = 1, ...)
```

`exGAUS()`

returns a `gamlss.family`

object which can be used to fit ex-Gaussian distribution in the `gamlss()`

function.

`dexGAUS()`

gives the density, `pexGAUS()`

gives the distribution function,

`qexGAUS()`

gives the quantile function, and `rexGAUS()`

generates random deviates.

- mu.link
Defines the

`mu.link`

, with "identity" link as the default for the`mu`

parameter.- sigma.link
Defines the

`sigma.link`

, with "log" link as the default for the`sigma`

parameter.- nu.link
Defines the

`nu.link`

, with "log" link as the default for the`nu`

parameter. Other links are "inverse", "identity", "logshifted" (shifted from one) and "own"- x,q
vector of quantiles

- mu
vector of

`mu`

parameter values- sigma
vector of scale parameter values

- nu
vector of

`nu`

parameter values- log, log.p
logical; if TRUE, probabilities p are given as log(p).

- lower.tail
logical; if TRUE (default), probabilities are P[X <= x], otherwise, P[X > x]

- p
vector of probabilities.

- n
number of observations. If

`length(n) > 1`

, the length is taken to be the number required- ...
for extra arguments

Mikis Stasinopoulos and Bob Rigby

The probability density function of the ex-Gaussian distribution, (`exGAUS`

), is defined as

$$f(y|\mu,\sigma,\nu)=\frac{1}{\nu} e^{\frac{\mu-y}{\nu}+\frac{\sigma^2}{2 \nu^2}} \Phi(\frac{y-\mu}{\sigma}-\frac{\sigma}{\nu})$$

where \(\Phi\) is the cdf of the standard normal distribution, for \(-\infty<y<\infty\), \(-\infty<\mu<\infty\), \(\sigma>0\) and \(\nu>0\) see pp. 372-373 of Rigby et al. (2019).

Cousineau, D. Brown, S. and Heathecote A. (2004) Fitting distributions using maximum likelihood: Methods and packages,
*Behavior Research Methods, Instruments and Computers*, **46**, 742-756.

Rigby, R. A. and Stasinopoulos D. M. (2005). Generalized additive models for location, scale and shape,(with discussion),
*Appl. Statist.*, **54**, part 3, pp 507-554.

Rigby, R. A., Stasinopoulos, D. M., Heller, G. Z., and De Bastiani, F. (2019) Distributions for modeling location, scale, and shape: Using GAMLSS in R, Chapman and Hall/CRC,tools:::Rd_expr_doi("10.1201/9780429298547"). An older version can be found in https://www.gamlss.com/.

Stasinopoulos D. M. Rigby R.A. (2007) Generalized additive models for location scale and shape (GAMLSS) in R.
*Journal of Statistical Software*, Vol. **23**, Issue 7, Dec 2007, tools:::Rd_expr_doi("10.18637/jss.v023.i07").

Stasinopoulos D. M., Rigby R.A., Heller G., Voudouris V., and De Bastiani F., (2017)
*Flexible Regression and Smoothing: Using GAMLSS in R*, Chapman and Hall/CRC. tools:::Rd_expr_doi("10.1201/b21973")

(see also https://www.gamlss.com/).

`gamlss.family`

, `BCCG`

, `GA`

,
`IG`

`LNO`

```
exGAUS() #
y<- rexGAUS(100, mu=300, nu=100, sigma=35)
hist(y)
# library(gamlss)
# m1<-gamlss(y~1, family=exGAUS)
# plot(m1)
curve(dexGAUS(x, mu=300 ,sigma=35,nu=100), 100, 600,
main = "The ex-GAUS density mu=300 ,sigma=35,nu=100")
plot(function(x) pexGAUS(x, mu=300,sigma=35,nu=100), 100, 600,
main = "The ex-GAUS cdf mu=300, sigma=35, nu=100")
```

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