Log normal distribution (from http://www.math.wm.edu/˜ leemis/chart/UDR/UDR.html ) The shorthand X ∼ log normal(α, β) is used to indicate that the random variable X has the log normal distribution with parameters α and β. A log normal random variable X with parameters α and β has probability density function

f(x) = \frac{1}{xβ\sqrt{2π}}e^{− \frac{1}{2}(ln(x/α)/β)^{2}} x > 0

for α and β > 0. The log normal distribution can be used to model the lifetime of an object, the weight of a person, or a service time. The central limit theorem indicates that the log normal distribution is useful for modeling random variables that can be thought of as a product of several independent random variables. The probability density function with three different parameter settings is illustrated below.

PSfrag replacements α = 1, β = 1

α = 1, β = 2
α = 5, β = 0.5
x
f(x)

The cumulative distribution function on the support of X is

F (x) = P(X ≤ x) = \frac{1}{2} + \frac{1}{2} erf( \frac{\sqrt{2}(ln(x)− α)}{2β} ) x > 0,

where

erf(x) = \frac{2}{\sqrt{π}} ∫_{0}^{x}e^{−t^{2}} dt.

The survivor function on the support of X is

S(x) = P(X ≥ x) = \frac{1}{2} − \frac{1}{2} erf( \frac{\sqrt{2}(ln(x)− α)}{2β} ) x > 0.

The hazard function on the support of X is

h(x) = \frac{f(x)}{S(x)} = −\sqrt{2}e^{−(ln(x)−α) ^{2}/2β^{2}} \frac{1}{\sqrt{π}} x^{−1} β^{−1} ( −1 + erf( \frac{\sqrt{2}(ln(x)− α)}{2β} ) ) ^{−1} x > 0.

The cumulative hazard function on the support of X is

H(x) = −lnS(x) = ln(2)+ iπ − ln( −1 + erf( \frac{\sqrt{2}(ln(x)− α)}{2β} ) ) x > 0.

The inverse distribution function, moment generating function, and characteristic function of X are mathematically intractable.

The median of X is α.

The population mean, variance, skewness, and kurtosis of X are

E[X] = αe^{β^{2}/2} V [X] = α^{2}e^{β^{2}}(e^{β^{2}} − 1)

E [( \frac{X − μ}{σ} ) ^{3}] = (e^{β^{2}} + 2)(e^{β^{2}} − 1)^{1/2} E [( \frac{X − μ}{σ} ) ^{4}] = e^{4β^{2}} + 2e^{3β^{2}} + 3e^{2β^{2}} − 3

APPL verification: The APPL statements

assume(alpha ¿ 0); assume(beta ¿ 0); X := [[x-¿ (1/(x*beta*sqrt(2*Pi)))*exp((-1/2)*(ln(x/alpha)/beta)ˆ 2)],

[0,infinity],[”Continuous”,”PDF”]]; CDF(X); SF(X); HF(X); CHF(X); Mean(X); Variance(X); Skewness(X); Kurtosis(X);

verify the cumulative distribution function, survivor function, hazard function, cumulative hazard function, population mean, variance, skewness, and kurtosis.