Editing Log-normal distribution (section)

=== Multivariate log-normal ===
If <math>\boldsymbol X \sim \mathcal{N}(\boldsymbol\mu,\,\boldsymbol\Sigma)</math> is a [[multivariate normal distribution]], then <math>Y_i = \exp(X_i)</math> has a multivariate log-normal distribution.<ref>{{Cite conference | last = Tarmast | first = Ghasem | year = 2001 | title = Multivariate Log–Normal Distribution | url = http://isi.cbs.nl/iamamember/CD2/pdf/329.PDF | archive-url = https://web.archive.org/web/20130719214220/http://isi.cbs.nl/iamamember/CD2/pdf/329.PDF | archive-date = 2013-07-19 | url-status = live | conference = ISI Proceedings: 53rd Session | location = Seoul}}</ref><ref>{{Cite conference | last = Halliwell | first = Leigh | year = 2015 | title = The Lognormal Random Multivariate | url = http://www.casact.org/pubs/forum/15spforum/Halliwell.pdf | archive-url = https://web.archive.org/web/20150930111908/http://www.casact.org/pubs/forum/15spforum/Halliwell.pdf | archive-date = 2015-09-30 | url-status = live | conference = Casualty Actuarial Society E-Forum, Spring 2015 | location = Arlington, VA}}</ref> The exponential is applied element-wise to the random vector <math>\boldsymbol X</math>. The mean of <math>\boldsymbol Y</math> is

<math display="block">\operatorname{E}[\boldsymbol Y]_i = e^{\mu_i + \frac{1}{2} \Sigma_{ii}} ,</math>

and its [[covariance matrix]] is

<math display="block">\operatorname{Var}[\boldsymbol Y]_{ij} = e^{\mu_i + \mu_j + \frac{1}{2}(\Sigma_{ii} + \Sigma_{jj}) } \left( e^{\Sigma_{ij}} - 1\right) . </math>

Since the multivariate log-normal distribution is not widely used, the rest of this entry only deals with the [[univariate distribution]].