Editing Log-normal distribution (section)

===Arithmetic moments===

For any real or complex number {{mvar|n}}, the {{mvar|n}}-th [[moment (mathematics)|moment]] of a log-normally distributed variable {{mvar|X}} is given by<ref name="JKB"/>
<math display="block">\operatorname{E}[X^n] = e^{n\mu + \frac{1}{2}n^2\sigma^2}.</math>

Specifically, the arithmetic mean, expected square, arithmetic variance, and arithmetic standard deviation of a log-normally distributed variable {{mvar|X}} are respectively given by:<ref name=":1" />

<math display="block">\begin{align}
 \operatorname{E}[X] & = e^{\mu + \tfrac{1}{2}\sigma^2}, \\[4pt]
 \operatorname{E}[X^2] & = e^{2\mu + 2\sigma^2}, \\[4pt]
 \operatorname{Var}[X] & = \operatorname{E}[X^2] - \operatorname{E}[X]^2
 = {\left(\operatorname{E}[X]\right)}^2 \left(e^{\sigma^2} - 1\right) \\[2pt]
 &= e^{2\mu + \sigma^2} \left(e^{\sigma^2} - 1\right), \\[4pt]
 \operatorname{SD}[X] & = \sqrt{\operatorname{Var}[X]}
 = \operatorname{E}[X] \sqrt{e^{\sigma^2} - 1} \\[2pt]
 &= e^{\mu + \tfrac{1}{2}\sigma^2} \sqrt{e^{\sigma^2} - 1},
\end{align}</math>

The arithmetic [[coefficient of variation]] <math>\operatorname{CV}[X]</math> is the ratio <math>\tfrac{\operatorname{SD}[X]}{\operatorname{E}[X]}</math>. For a log-normal distribution it is equal to<ref name=":2" />
<math display="block">\operatorname{CV}[X] = \sqrt{e^{\sigma^2} - 1}.</math>
This estimate is sometimes referred to as the "geometric CV" (GCV),<ref>Sawant, S.; Mohan, N. (2011) [http://pharmasug.org/proceedings/2011/PO/PharmaSUG-2011-PO08.pdf "FAQ: Issues with Efficacy Analysis of Clinical Trial Data Using SAS"] {{webarchive | url = https://web.archive.org/web/20110824094357/http://pharmasug.org/proceedings/2011/PO/PharmaSUG-2011-PO08.pdf | date = 24 August 2011 }}, ''PharmaSUG2011'', Paper PO08</ref><ref>{{cite journal | last1 = Schiff | first1 = MH | display-authors = etal | year = 2014 | title = Head-to-head, randomised, crossover study of oral versus subcutaneous methotrexate in patients with rheumatoid arthritis: drug-exposure limitations of oral methotrexate at doses >=15 mg may be overcome with subcutaneous administration | journal = Ann Rheum Dis | volume = 73 | issue = 8 | pages = 1–3 | doi = 10.1136/annrheumdis-2014-205228 | pmid = 24728329 | pmc = 4112421}}</ref> due to its use of the geometric variance. Contrary to the arithmetic standard deviation, the arithmetic coefficient of variation is independent of the arithmetic mean.

The parameters {{math|''μ''}} and {{math|''σ''}} can be obtained, if the arithmetic mean and the arithmetic variance are known:

<math display="block">\begin{align}
\mu &= \ln \frac{\operatorname{E}[X]^2}{\sqrt{\operatorname{E}[X^2]}}
= \ln \frac{\operatorname{E}[X]^2}{\sqrt{\operatorname{Var}[X] + \operatorname{E}[X]^2}}, \\[1ex]
 \sigma^2 &= \ln \frac{\operatorname{E}[X^2]}{\operatorname{E}[X]^2}
= \ln \left(1 + \frac{\operatorname{Var}[X]}{\operatorname{E}[X]^2}\right).
 \end{align}</math>

A probability distribution is not uniquely determined by the moments {{math|1=E[''X''<sup>''n''</sup>] = e<sup>''nμ'' + {{sfrac|1|2}}''n''<sup>2</sup>''σ''<sup>2</sup></sup>}} for {{math|''n'' ≥ 1}}. That is, there exist other distributions with the same set of moments.<ref name="JKB"/> In fact, there is a whole family of distributions with the same moments as the log-normal distribution.{{Citation needed|date=March 2012}}