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In probability theory, a log-normal (or lognormal) distribution is a continuous probability distribution of a random variable whose logarithm is normally distributed. Thus, if the random variable X is log-normally distributed, then Y = ln(X) has a normal distribution. Equivalently, if Y has a normal distribution, then the exponential function of Y, X = exp(Y) , has a log-normal distribution. A random variable which is log-normally distributed takes only positive real values. It is a convenient and useful model for measurements in exact and engineering sciences, as well as medicine, economics and other topics (e.g., energies, concentrations, lengths, prices of financial instruments, and other metrics). The distribution is occasionally referred to as the Galton distribution or Galton's distribution, after Francis Galton. The log-normal distribution has also been associated with other names, such as McAlister, Gibrat and Cobb–Douglas.A log-normal process is the statistical realization of the multiplicative product of many independent random variables, each of which is positive. This is justified by considering the central limit theorem in the log domain (sometimes called Gibrat's law). The log-normal distribution is the maximum entropy probability distribution for a random variate X—for which the mean and variance of ln(X) are specified. Definitions Generation and parameters Let  Z {\displaystyle \ Z\ } be a standard normal variable, and let μ{\displaystyle \mu } and σ{\displaystyle \sigma } be two real numbers, with σ>0{\displaystyle \sigma >0}. Then, the distribution of the random variable X=eμ+σZ{\displaystyle X=e^{\mu +\sigma Z}}is called the log-normal distribution with parameters μ{\displaystyle \mu } and σ{\displaystyle \sigma }. These are the expected value (or mean) and standard deviation of the variable's natural logarithm, not the expectation and standard deviation of  X {\displaystyle \ X\ } itself. This relationship is true regardless of the base of the logarithmic or exponential function: If  loga⁡(X) {\displaystyle \ \log _{a}(X)\ } is normally distributed, then so is  logb⁡(X) {\displaystyle \ \log _{b}(X)\ } for any two positive numbers  a,b≠1 .{\displaystyle \ a,b\neq 1~.} Likewise, if  eY {\displaystyle \ e^{Y}\ } is log-normally distributed, then so is  aY ,{\displaystyle \ a^{Y}\ ,} where 0<a≠1{\displaystyle 0<a\neq 1}. In order to produce a distribution with desired mean μX{\displaystyle \mu _{X}} and variance  σX2 ,{\displaystyle \ \sigma _{X}^{2}\ ,} one uses  μ=ln⁡(μX2 μX2+σX2  ) {\displaystyle \ \mu =\ln \left({\frac {\mu _{X}^{2}}{\ {\sqrt {\mu _{X}^{2}+\sigma _{X}^{2}\ }}\ }}\right)\ } and  σ2=ln⁡(1+ σX2 μX2) .{\displaystyle \ \sigma ^{2}=\ln \left(1+{\frac {\ \sigma _{X}^{2}\ }{\mu _{X}^{2}}}\right)~.} Alternatively, the "multiplicative" or "geometric" parameters  μ∗=eμ {\displaystyle \ \mu ^{*}=e^{\mu }\ } and  σ∗=eσ {\displaystyle \ \sigma ^{*}=e^{\sigma }\ } can be used. They have a more direct interpretation:  μ∗ {\displaystyle \ \mu ^{*}\ } is the median of the distribution, and  σ∗ {\displaystyle \ \sigma ^{*}\ } is useful for determining "scatter" intervals, see below. Probability density function A positive random variable  X {\displaystyle \ X\ } is log-normally distributed (i.e.,  X∼Lognormal⁡( μx,σx2 ) {\displaystyle \ X\sim \operatorname {Lognormal} \left(\ \mu _{x},\sigma _{x}^{2}\ \right)\ }), if the natural logarithm of  X {\displaystyle \ X\ } is normally distributed with mean μ{\displaystyle \mu } and variance  σ2 :{\displaystyle \ \sigma ^{2}\ :} ln⁡(X)∼N(μ,σ2){\displaystyle \ln(X)\sim {\mathcal {N}}(\mu ,\sigma ^{2})}Let  Φ {\displaystyle \ \Phi \ } and  φ {\displaystyle \ \varphi \ } be respectively the cumulative probability distribution function and the probability density function of the  N( 0,1 ) {\displaystyle \ {\mathcal {N}}(\ 0,1\ )\ } standard normal distribution, then we have that the probability density function of the log-normal distribution is given by: Cumulative distribution function The cumulative distribution function is FX(x)=Φ((ln⁡x)−μσ){\displaystyle F_{X}(x)=\Phi \left({\frac {(\ln x)-\mu }{\sigma }}\right)}where  Φ {\displaystyle \ \Phi \ } is the cumulative distribution function of the standard normal distribution (i.e.,  N⁡( 0, 1) {\displaystyle \ \operatorname {\mathcal {N}} (\ 0,\ 1)\ }). This may also be expressed as follows: 12[1+erf⁡(ln⁡x−μσ2)]=12erfc⁡(−ln⁡x−μσ2){\displaystyle {\frac {1}{2}}\left[1+\operatorname {erf} \left({\frac {\ln x-\mu }{\sigma {\sqrt {2}}}}\right)\right]={\frac {1}{2}}\operatorname {erfc} \left(-{\frac {\ln x-\mu }{\sigma {\sqrt {2}}}}\right)}where erfc is the complementary error function. Multivariate log-normal If X∼N(μ,Σ){\displaystyle {\boldsymbol {X}}\sim {\mathcal {N}}({\boldsymbol {\mu }},\,{\boldsymbol {\Sigma }})} is a multivariate normal distribution, then Yi=exp⁡(Xi){\displaystyle Y_{i}=\exp(X_{i})} has a multivariate log-normal distribution. The exponential is applied elementwise to the random vector X{\displaystyle {\boldsymbol {X}}}. The mean of Y{\displaystyle {\boldsymbol {Y}}} is E⁡[Y]i=eμi+12Σii,{\displaystyle \operatorname {E} [{\boldsymbol {Y}}]_{i}=e^{\mu _{i}+{\frac {1}{2}}\Sigma _{ii}},}and its covariance matrix is Var⁡[Y]ij=eμi+μj+12(Σii+Σjj)(eΣij−1).{\displaystyle \operatorname {Var} [{\boldsymbol {Y}}]_{ij}=e^{\mu _{i}+\mu _{j}+{\frac {1}{2}}(\Sigma _{ii}+\Sigma _{jj})}(e^{\Sigma _{ij}}-1).}Since the multivariate log-normal distribution is not widely used, the rest of this entry only deals with the univariate distribution. Characteristic function and moment generating function All moments of the log-normal distribution exist and E⁡[Xn]=enμ+n2σ2/2{\displaystyle \operatorname {E} [X^{n}]=e^{n\mu +n^{2}\sigma ^{2}/2}}This can be derived by letting z=ln⁡(x)−(μ+nσ2)σ{\displaystyle z={\tfrac {\ln(x)-(\mu +n\sigma ^{2})}{\sigma }}} within the integral. However, the log-normal distribution is not determined by its moments. This implies that it cannot have a defined moment generating function in a neighborhood of zero. Indeed, the expected value E⁡[etX]{\displaystyle \operatorname {E} [e^{tX}]} is not defined for any positive value of the argument t{\displaystyle t}, since the defining integral diverges. The characteristic function E⁡[eitX]{\displaystyle \operatorname {E} [e^{itX}]} is defined for real values of t, but is not defined for any complex value of t that has a negative imaginary part, and hence the characteristic function is not analytic at the origin. Consequently, the characteristic function of the log-normal distribution cannot be represented as an infinite convergent series. In particular, its Taylor formal series diverges: ∑n=0∞(it)nn!enμ+n2σ2/2{\displaystyle \sum _{n=0}^{\infty }{\frac {(it)^{n}}{n!}}e^{n\mu +n^{2}\sigma ^{2}/2}}However, a number of alternative divergent series representations have been obtained.A closed-form formula for the characteristic function φ(t){\displaystyle \varphi (t)} with t{\displa.... Discover the Paul Limpert popular books. Find the top 100 most popular Paul Limpert books.