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# Probability distribution summary

This page summarize the distribution available in Mlxtran and their use.

### Normal distribution and the associated transformed.

The first class of distribution is the normal distribution and the associated transformation. Then, we have

• Normal distribution, used with keyword normal (h(X)=X)
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=1}
• Logit-normal distribution, used with keyword logitnormal ( $h(X)=\frac{1}{1+e^{-X}}$)
y_lgn = {distribution=logitnormal, typical=0, sd=1}
• Probit-normal distribution, used with keyword probitnormal ( $h(X)=\Psi^{-1}(X)$, where $\Psi$ is the cumulative distribution function of the ${\cal N}(0, 1)$ distribution)
y_pbn = {distribution=probitnormal, typical=.5, sd=1}

Notice that to clarify the use between mean and typical, additional keywords

• meanlog, meanlogit, and meanProbit can be used to replace mean for the log-normal, the logit-normal and the probit-normal distribution respectively.
• sdlog, sdlogit, and sdprobit can be used to replace sd for the log-normal, the logit-normal and the probit-normal distribution respectively.

This leads to a possibility to define a distribution as follows

• Log-normal distribution,
y_ln = {distribution=lognormal, meanlog=1, sdlog=1}
• Logit-normal distribution,
y_lgn = {distribution=logitnormal, meanlogit=.5, sdlogit=1}
• Probit-normal distribution,
y_pbn = {distribution=probitnormal, meanProbit=0.5, sdProbit=1}

These distributions can be used in Mlxplore, Monolix, and Simulx.

### Continuous distribution available for simulation.

These are the following additional distribution available for simulation purpose.

• 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}

These distributions can be used in Mlxplore and Simulx.

### Discrete distribution available for simulation.

These are the following additional distribution available for simulation purpose

• 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}

These distributions can be used only in Simulx, as Mlxplore can not manage discrete data.