Lognormal Distribution

The lognormal probability density function is often used to define material properties.

See Also
About Probability Distributions

Given a random variable X defined over 0<x<, and given that Y=1nX is normally distributed with mean μY and standard deviation σY, the random variable X follows the Lognormal distribution, defined by the probability density function.

The probability density function is

fX(x)=1βx2πexp[12β2(logxα)2]x>0=0otherwise.

The lognormal distribution function is

FX(x)=Φ(1nxαβ),

where α=μY and β=σY. The mean and standard deviation of the random variable X are given as follows:

μX=e(α+12β2)

and

σX=μX2(eβ21).

The lognormal probability density function, as shown in the figure below, is often used to describe material properties, sizes from a breakage process, and the life of some types of transistors.