Probability distribution summary

This page summarizes the distributions available in Mlxtran. They can be used to define random variables in the [INDIVIDUAL] section (individual parameters) or in the [LONGITUDINAL] section (observations).

Normal distribution and the associated transformed – Monolix and Simulx

The normal distribution and its transformations can be used in Monolix and Simulx. The user can either define the mean of the associated normal distribution or the typical value of the distribution; as well as the standard deviation sd or variance var of the associated normal distribution. For the logitnormal distribution, the min and max can also be specified (optional). Covariates and correlations between random effects can also be added, see the [INDIVIDUAL] section.

Normal distribution, used with keyword normal: \(h(X)=X \)

y_norm = {distribution=normal, typical=0, sd=1}
y_norm = {distribution=normal, mean=0, sd=1}

Log-normal distribution, used with keyword lognormal: \(h(X)=\log(X) \)

y_ln = {distribution=lognormal, typical=1, sd=0.3} 
y_ln = {distribution=lognormal, mean=0, sd=0.3}

Logit-normal distribution, used with keyword logitnormal: \(h(X)=\log(\frac{X-a}{b-X}) \) with \(a\) and \(b\) the bounds of the logit-normal distribution indicated in the optional arguments min and max. By default, \(a=0\) and \(b=1\).
```
y_lgn = {distribution=logitnormal, typical=0.5, sd=0.6, min=0, max=1} 
y_lgn = {distribution=logitnormal, mean=0, sd=0.6, min=0, max=1}
```
Probit-normal distribution, used with keyword probitnormal: \(h(X)=\Phi^{-1}(X) \), where \(\Phi \) is the cumulative distribution function of the standard normal distribution
```
y_pbn = {distribution=probitnormal, typical=0.5, sd=0.6} 
y_pbn = {distribution=probitnormal, mean=0, sd=0.6}
```

Other distributions – Simulx only

Additional distributions are available for simulation purpose in Simulx. They include continuous and discrete distributions. These distributions can neither be combined with the argument covariate, nor with the definition of correlations of random effects.

Continuous distributions

Uniform distribution, used with keyword uniform,
```
y_uni = {distribution=uniform, min=0, max=100}
```

Exponential distribution, used with keyword exponential,
```
y_exp = {distribution=exponential, rate=0.7}
```

Gamma distribution, used with keyword gamma,

y_gamma = {distribution=gamma, shape=2, scale=0.5}

Weibull distribution, used with keyword weibull,

y_weibull = {distribution=weibull, shape=0.75, scale=1.2}

ExtremeValue distribution, used with keyword extremeValue,

y_ev = {distribution=extremeValue, location=-10, scale=1.8}

ChiSquared distribution, used with keyword chiSquared,
```
y_cs = {distribution=chiSquared, df=2}
```

Cauchy distribution, used with keyword cauchy,

y_cauchy = {distribution=cauchy, location=-5.2, scale=2.7}

FisherF distribution, used with keyword fisherF,

y_fisherF = {distribution=fisherF, df1=10, df2=9}

StudentT distribution, used with keyword studentT,
```
y_studentT = {distribution=studentT, df=2}
```

Discrete distributions

Bernoulli distribution, used with keyword bernoulli,
```
y_ber = {distribution=bernoulli, prob=0.3}
```
Discrete uniform distribution, used with keyword discreteUniform,
```
y_du = {distribution=discreteUniform, min=-4, max=2}
```

Binomial distribution, used with keyword binomial,

y_bi = {distribution=binomial, size=30, prob=0.7}

Geometric distribution, used with keyword geometric,
```
y_ber = {distribution=geometric, prob=0.2}
```
Negative binomial distribution, used with keyword negativeBinomial,
```
y_nb = {distribution=negativeBinomial, size=3, prob=0.1}
```
Poisson distribution, used with keyword poisson,
```
y_nb = {distribution=poisson, lambda=0.1}
```