Lognormal Distribution

Given a random variable $X$ defined over $0 < x < \infty$ , and given that $Y = 1 n X$ is normally distributed with mean $μ_{Y}$ and standard deviation $σ_{Y}$ , the random variable $X$ follows the Lognormal distribution, defined by the probability density function.

The probability density function is

\begin{array}{l} f_{X} (x) & = \frac{1}{β (x - t) \sqrt{2 π}} \exp [- \frac{1}{2 β^{2}} {(\log (x - t) - α)}^{2}]; & x > 0, 0 \leq t < x \\ = 0; & x \leq 0, t > x \end{array}

The lognormal distribution function is

F_{X} (x) = Φ (\frac{1 n (x - t) - α}{β})

Note that $α = μ_{Y}$ and $β = σ_{Y}$ . The mean and standard deviation of the random variable $X$ are given as follows:

μ_{X} = e^{(α + \frac{1}{2} β^{2})} + t

and

σ_{X} = \sqrt{μ_{(X - t)}^{2} (e^{β^{2}} - 1)} .

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