Each of the distributions is described in a two-page summary. The summary header
includes the distribution name and the parameter list, along with the numerical range for
which variates and parameters (if constrained) are defined.
Each distribution is illustrated with at least one example. In this figure, the parameters
used are shown in parentheses, in the order listed in the header. Expressions are then given
for the PDF and CDF. Remaining subsections, as appropriate, are as follows:
Parameters
This is an interpretation of the meaning of each parameter, with the usual literature
symbol (if any) given in parentheses.
Unless otherwise indicated, parameter A is a location parameter, positioning the
overall distribution along the abscissa. Parameter B is a scale parameter, describing
the extent of the distribution. Parameters C and, possibly, D are shape parameters
which affect skewness, kurtosis, etc. In the case of binary mitures, there is also a
weight, p, for the first component.
Moments, etc.
Provided that there are closed forms, the mean, variance, skewness, kurtosis, mode,
median, first quartile (Q1), and third quartile (Q3) are described along with the
quantiles for the mean (qMean) and mode (qMode). If random variates are
computable with a closed-form expression, the latter is also given.
Note that the mean and variance have their usual units while the skewness and
kurtosis are dimensionless. Furthermore, the kurtosis is referenced to that of a
standard Normal distribution (kurtosis = 3).
Notes
These include any relevant constraints, cautions, etc.
Aliases and Special Cases
These alternate names are also listed as well in the Table of Contents.
Characterizations
This list is far from exhaustive. It is intended simply to convey a few of the more
important situations in which the distribution is particularly relevant.
Obviously, so brief a account cannot begin to do justice to the wealth of information
available. For fuller accounts, the aforementioned references, [KOT82], [JOH92], and
[JOH94] are excellent starting points.