1Blurb:: 2Aleatory uncertain variable - lognormal 3 4Description:: 5If the logarithm of an uncertain variable X has a normal distribution, that is 6\f$\log X \sim N(\mu,\sigma)\f$, 7then X is distributed with a lognormal distribution. 8The lognormal is often used to model: 9-# time to perform some task 10-# variables which are the product of a large number of other quantities, by the Central Limit Theorem 11-# quantities which cannot have negative values. 12 13The number of lognormal uncertain variables, their means, and either 14standard deviations or error factors must be specified, while the 15distribution lower and upper bounds and variable descriptors are 16optional specifications. These distribution bounds can be used to 17truncate the tails of lognormal distributions, which as for bounded 18normal, can result in the mean and the standard deviation of the 19sample data being different from the mean and standard deviation of 20the underlying distribution (see "bounded lognormal" and "bounded 21lognormal-n" distribution types in \cite Wyss1998). 22 23For the lognormal variables, one may specify either the mean 24\f$\mu\f$ 25and standard deviation 26\f$\sigma\f$ 27of the actual lognormal distribution (option 1), the mean 28\f$\mu\f$ 29 and error factor 30\f$\epsilon\f$ 31of the actual lognormal distribution (option 2), or the mean 32\f$\lambda\f$ ("lambda") 33and standard deviation 34\f$\zeta\f$ ("zeta") 35of the underlying normal distribution (option 3). 36<!-- This approach diverges from that of \cite Wyss1998, which assumes that a specification of mean and standard deviation provides parameters of the underlying normal distribution, and a specification of mean and error factor provides statistics of the actual lognormal distribution. --> 37 38The conversion equations from lognormal mean 39\f$\mu\f$ and either lognormal error factor 40\f$\epsilon\f$ 41or lognormal standard deviation 42\f$\sigma\f$ 43to the mean 44\f$\lambda\f$ 45and standard deviation 46\f$\zeta\f$ 47of the underlying normal distribution are as follows: 48 49\f[\zeta = \frac{\ln(\epsilon)}{1.645}\f] 50\f[\zeta^2 = \ln\left(\frac{\sigma^2}{\mu^2} + 1\right)\f] 51\f[\lambda = \ln(\mu) - \frac{\zeta^2}{2}\f] 52 53Conversions from \f$\lambda\f$ and \f$\zeta\f$ back to \f$\mu\f$ 54and \f$\epsilon\f$ or \f$\sigma\f$ are as follows: 55 56\f[\mu = \exp \left( \lambda + \frac{\zeta^2}{2} \right) \f] 57\f[\sigma^2 = \exp \left( 2\lambda + \zeta^2 \right) \left( \exp \left(\zeta^2\right) - 1\right)\f] 58\f[\epsilon = \exp \left( 1.645\zeta \right) \f] 59 60The density function for the lognormal distribution is: 61\f[f(x) = \frac{1}{\sqrt{2\pi}\zeta x} 62 \exp \left( -\frac{1}{2}\left(\frac{\ln(x)-\lambda}{\zeta}\right)^2 \right) \f] 63 64Topics:: continuous_variables, aleatory_uncertain_variables 65Examples:: 66Theory:: 67When used with some methods such as design of experiments and 68multidimensional parameter studies, distribution bounds are inferred 69to be [0, \f$\mu + 3 \sigma\f$]. 70 71For some methods, including vector and centered parameter studies, an 72initial point is needed for the uncertain variables. When not given 73explicitly, these variables are initialized to their means, correcting 74to bounds if needed. 75 76Faq:: 77See_Also:: 78