1. Exploratory Data Analysis 1.3. EDA Techniques 1.3.6. Probability Distributions 1.3.6.6. Gallery of Distributions 1.3.6.6.9. Lognormal Distribution 

Probability Density Function 
A variable X is lognormally distributed if \(Y = \ln(X)\) is
normally distributed with "LN" denoting the natural logarithm.
The general formula for the probability
density function of the lognormal distribution is
\( f(x) = \frac{e^{((\ln((x\theta)/m))^{2}/(2\sigma^{2}))}} {(x\theta)\sigma\sqrt{2\pi}} \hspace{.2in} x > \theta; m, \sigma > 0 \) where σ is the shape parameter (and is the standard deviation of the log of the distribution), θ is the location parameter and m is the scale parameter (and is also the median of the distribution). If x = θ, then f(x) = 0. The case where θ = 0 and m = 1 is called the standard lognormal distribution. The case where θ equals zero is called the 2parameter lognormal distribution. The equation for the standard lognormal distribution is \( f(x) = \frac{e^{((\ln x)^{2}/2\sigma^{2})}} {x\sigma\sqrt{2\pi}} \hspace{.2in} x > 0; \sigma > 0 \) Since the general form of probability functions can be expressed in terms of the standard distribution, all subsequent formulas in this section are given for the standard form of the function. Note that the lognormal distribution is commonly parameterized with \( \mu = \log(m) \) The μ parameter is the mean of the log of the distribution. If the μ parameterization is used, the lognormal pdf is \( f(x) = \frac{e^{(\ln(x  \theta)  \mu)^2/(2\sigma^2)}} {(x  \theta)\sigma\sqrt{2\pi}} \hspace{.2in} x > \theta; \sigma > 0 \) We prefer to use the m parameterization since m is an explicit scale parameter. The following is the plot of the lognormal probability density function for four values of σ.
There are several common parameterizations of the lognormal distribution. The form given here is from Evans, Hastings, and Peacock. 

Cumulative Distribution Function 
The formula for the cumulative distribution
function of the lognormal distribution is
\( F(x) = \Phi(\frac{\ln(x)} {\sigma}) \hspace{.2in} x \ge 0; \sigma > 0 \) where \(\Phi\) is the cumulative distribution function of the normal distribution. The following is the plot of the lognormal cumulative distribution function with the same values of σ as the pdf plots above.


Percent Point Function 
The formula for the percent point
function of the lognormal distribution is
\( G(p) = \exp(\sigma\Phi^{1}(p)) \hspace{.2in} 0 \le p < 1; \sigma > 0 \) where \(\Phi^{1}\) is the percent point function of the normal distribution. The following is the plot of the lognormal percent point function with the same values of σ as the pdf plots above.


Hazard Function 
The formula for the hazard
function of the lognormal distribution is
\( h(x,\sigma) = \frac{(\frac{1} {x\sigma})\phi(\frac{\ln x} {\sigma})} {\Phi(\frac{\ln x} {\sigma})} \hspace{.2in} x > 0; \sigma > 0 \) where \(\phi\) is the probability density function of the normal distribution and \(\Phi\) is the cumulative distribution function of the normal distribution. The following is the plot of the lognormal hazard function with the same values of σ as the pdf plots above.


Cumulative Hazard Function 
The formula for the cumulative
hazard function of the lognormal distribution is
\( H(x) = \ln(1  \Phi(\frac{\ln(x)} {\sigma})) \hspace{.2in} x \ge 0; \sigma > 0 \) where \(\Phi\) is the cumulative distribution function of the normal distribution. The following is the plot of the lognormal cumulative hazard function with the same values of σ as the pdf plots above.


Survival Function 
The formula for the survival function
of the lognormal distribution is
\( S(x) = 1  \Phi(\frac{\ln(x)} {\sigma}) \hspace{.2in} x \ge 0; \sigma > 0 \) where \(\Phi\) is the cumulative distribution function of the normal distribution. The following is the plot of the lognormal survival function with the same values of σ as the pdf plots above.


Inverse Survival Function 
The formula for the inverse
survival function of the lognormal distribution is
\( Z(p) = \exp(\sigma\Phi^{1}(1p)) \hspace{.2in} 0 \le p < 1; \sigma > 0 \) where \(\Phi^{1}\) is the percent point function of the normal distribution. The following is the plot of the lognormal inverse survival function with the same values of σ as the pdf plots above.


Common Statistics 
The formulas below are with the location parameter equal to zero
and the scale parameter equal to one.


Parameter Estimation 
The maximum likelihood estimates for the scale parameter,
m, and the shape parameter, σ, are


Comments  The lognormal distribution is used extensively in reliability applications to model failure times. The lognormal and Weibull distributions are probably the most commonly used distributions in reliability applications.  
Software  Most general purpose statistical software programs support at least some of the probability functions for the lognormal distribution. 