Log-normal distribution

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. Likewise, 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. The distribution is occasionally referred to as the Galton distribution or Galton's distribution, after Francis Galton. The log-normal distribution also has been associated with other names, such as McAlister, Gibrat and Cobb–Douglas. 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. Likewise, 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. The distribution is occasionally referred to as the Galton distribution or Galton's distribution, after Francis Galton. The log-normal distribution also has 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. 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. Given a log-normally distributed random variable X {displaystyle X} and two parameters μ {displaystyle mu } and σ {displaystyle sigma } that are, respectively, the mean and standard deviation of the variable’s natural logarithm, then the logarithm of X {displaystyle X} is normally distributed, and we can write X {displaystyle X} as with Z {displaystyle Z} a standard normal variable. This relationship is true regardless of the base of the logarithmic or exponential function. If log a ⁡ ( Y ) {displaystyle log _{a}(Y)} is normally distributed, then so is log b ⁡ ( Y ) {displaystyle log _{b}(Y)} , for any two positive numbers a , b ≠ 1 {displaystyle a,b eq 1} . Likewise, if e X {displaystyle e^{X}} is log-normally distributed, then so is a X {displaystyle a^{X}} , where 0 < a ≠ 1 {displaystyle 0<a eq 1} . The two parameters μ {displaystyle mu } and σ {displaystyle sigma } are not location and scale parameters for a lognormally distributed random variable X, but they are respectively location and scale parameters for the normally distributed logarithm ln(X). The quantity eμ is a scale parameter for the family of lognormal distributions. In contrast, the mean and variance of the non-logarithmized sample values are respectively denoted m {displaystyle m} , and v {displaystyle v} in this article. The two sets of parameters can be related as (see also Arithmetic moments below) A positive random variable X is log-normally distributed if the logarithm of X is normally distributed, 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) distribution.