Other PDFs
Author: John M. Cimbala, Penn State University
Latest revision: 30 January 2008
Introduction
•
There are
other
useful standard distributions and PDFs besides the Gaussian PDF. These include the
binomial, chi-squared, exponential, gamma, lognormal, Poisson, student’s
t
, uniform, and Weibull PDFs.
•
We discuss some of these in this learning module, although not in as much detail as for the Gaussian
(normal) distribution.
Lognormal PDF
•
A
lognormal PDF
is defined as
a PDF that becomes Gaussian when the x-axis is plotted as a log scale
.
o
Lognormal PDFs often appear in air quality measurements, e.g., the size distribution of particles. It is
also useful for some life and durability analyses of components and equipment or instruments.
o
When the PDF is plotted as usual (linear
x
scale), it is skewed towards the left (lower values), and has a
very long tail to the right (higher values). This is shown on the first plot below.
o
However, when the PDF is plotted with a logarithmic
x
scale, all else being equal, it is no longer skewed,
but becomes symmetric. In fact, it’s bell shape is identical to that of a Gaussian or normal PDF. This is
shown on the second plot below.
o
Another way to plot lognormal PDFs is to first convert the
x
values to log
10
(
x
) or ln(
x
), and then plot
using a
linear
abscissa scale. Either way, the PDF again looks like a standard Gaussian PDF, as
illustrated below.
o
To calculate statistics with a lognormal PDF, we substitute either log
10
(
x
) or ln(
x
) as our variable instead
of
x
itself. For example, if the data are for particle diameter
D
p
in units of microns (
μ
m), we let our
statistics variable be
x
= ln[
D
p
/(1
μ
m)] instead of
D
p
itself. All statistics are then based on
x
as usual.