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.