Editing Log-normal distribution (section)

==== Examples for re-parameterization ====
Consider the situation when one would like to run a model using two different optimal design tools, for example PFIM<ref>{{cite journal | last1 = Retout | first1 = S | last2 = Duffull | first2 = S | last3 = Mentré | first3 = F | year = 2001 | title = Development and implementation of the population Fisher information matrix for the evaluation of population pharmacokinetic designs | journal = Comp Meth Pro Biomed | volume = 65 | issue = 2 | pages = 141–151 | doi = 10.1016/S0169-2607(00)00117-6 | pmid = 11275334 }}</ref> and PopED.<ref>The PopED Development Team (2016). PopED Manual, Release version 2.13. Technical report, Uppsala University.</ref> The former supports the LN2, the latter LN7 parameterization, respectively. Therefore, the re-parameterization is required, otherwise the two tools would produce different results.

For the transition <math>\operatorname{LN2}(\mu, v) \to \operatorname{LN7}(\mu_N, \sigma_N)</math> following formulas hold <math display="inline">\mu_N = \exp(\mu+v/2) </math> and <math display="inline">\sigma_N = \exp(\mu+v/2)\sqrt{\exp(v)-1}</math>.

For the transition <math>\operatorname{LN7}(\mu_N, \sigma_N) \to \operatorname{LN2}(\mu, v)</math> following formulas hold <math display="inline">\mu = \ln \mu_N - \frac{1}{2} v </math> and <math display="inline"> v = \ln(1+\sigma_N^2/\mu_N^2)</math>.

All remaining re-parameterisation formulas can be found in the specification document on the project website.<ref name="probontoWebsite">ProbOnto website, URL: http://probonto.org</ref>