Lognormal distribution in Base e | Vose Software

# Lognormal distribution in Base e

Format: LognormalE(m, s)

The definition of a Lognormally distributed random variable is that the log of that variable is Normally distributed. Thus, an alternative way of defining a Lognormally distributed variable is to specify the mean and standard deviation of the corresponding Normal distribution. One usually defines the Lognormal distribution in terms of its mean and standard deviation, as in the Lognormal distribution. However, the LognormalE distribution uses the natural exponent e as the base:

EXP(Normal(m,s)) = LognormalE(m,s)

## Uses

Taking logs might seem like an unnecessary complication, but many people whose work involves statistics and who consider Lognormal random variables, habitually record and discuss the mean and the standard deviation of a corresponding Normal distribution. For example stock prices are frequently modeled using Geometric Brownian Motion which assumes that the price S varies in some small amount of time dt by an amount dS as follows:

Integrating over time, the stock price St after time t is related to its initial value S0 by:

## Alternative parameterizations

The Lognormal distribution has as its parameters the mean and standard deviation of the distribution itself.

The LognormalB distribution has as its parameters the mean and standard deviation of the Normal distribution of logB[X].

## ModelRisk functions added to Microsoft Excel for the Lognormal distribution in base e

VoseLogNormalE generates random values from this distribution for Monte Carlo simulation, or calculates a percentile if used with a U parameter.

VoseLogNormalEObject constructs a distribution object for this distribution.

VoseLogNormalEProb returns the probability density or cumulative distribution function for this distribution.

VoseLogNormalEProb10 returns the log10 of the probability density or cumulative distribution function.

VoseLogNormalEFit generates values from this distribution fitted to data, or calculates a percentile from the fitted distribution..

VoseLogNormalEFitObject constructs a distribution object of this distribution fitted to data.

VoseLogNormalEFitP returns the parameters of this distribution fitted to data.