1.3. EDA Techniques
1.3.6. Probability Distributions
1.3.6.6. Gallery of Distributions
1.3.6.6.9. Lognormal Distribution
where 
is the
shape parameter,
is the
location parameter and
m is the scale parameter.
The case where = 0 and
m = 1 is called the standard lognormal
distribution. The case where
equals zero
is called the 2-parameter lognormal distribution.
The equation for the standard lognormal distribution is
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.
The following is the plot of the lognormal probability density
function for four values of .
There are several common parameterizations of the lognormal distribution. The form given here is from Evans, Hastings, and Peacock.
where 
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
as the pdf plots above.
where 
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 as the pdf
plots above.
where 
is the
probability density function of the
normal distribution and is the
cumulative distribution function of the
normal distribution.
The following is the plot of the lognormal hazard function with the
same values of as the pdf plots above.
where 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 as the
pdf plots above.
where is the
cumulative distribution function of the
normal distribution.
The following is the plot of the lognormal survival function
with the same values of as the
pdf plots above.
where 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 as the
pdf plots above.
| Mean |
|
| Median | Scale parameter m (= 1 if scale parameter not specified). |
| Mode |
|
| Range | Zero to positive infinity |
| Standard Deviation |
|
| Skewness |
|
| Kurtosis |
|
| Coefficient of Variation |
|
, are
![sigmahat = SQRT{SUM[i=1 to N][(LOG(X(i))-mu)**2]/N}](eqns/sigmahat.gif)
![Uhat = SUM[i=1 to N][LOG(X(i))]/N](eqns/uhat.gif)
