Generates random variates from the gamma distribution, and optionally, illustrates
the use of the inverse-cdf technique,
the effect of random sampling variability in relation to the pdf and cdf.
When all of the graphics are requested,
the top graph illustrates the use of the inverse-cdf technique by
graphing the population cdf and the transformation of the random numbers
to random variates,
the middle graph illustrates the effect of random sampling variability
by graphing the population pdf and the histogram associated with the
random variates, and
the bottom graph illustrates effect of random sampling variability by
graphing the population cdf and the empirical cdf associated with the
random variates.
All aspects of the random variate generation algorithm are output in red. All
aspects of the population distribution are output in black.
The gamma distribution with parameters shape = \(a\) and scale
= \(s\) has density
$$f(x) = \frac{1}{s^a\, \Gamma(a)} x^{a-1} e^{-x/s}$$
for \(x \ge 0\), \(a > 0\), and \(s > 0\). (Here \(\Gamma(a)\)
is the function implemented by R's gamma() and
defined in its help.)
The population mean and variance are \(E(X) = as\) and \(Var(X) = as^2\).
The algorithm for generating random variates from the gamma distribution is
synchronized (one random variate for each random number) and monotone in u.
This means that the variates generated here might be useful in some variance
reduction techniques used in Monte Carlo and discrete-event simulation.
Values from the u vector are plotted in the cdf plot along the vertical axis
as red dots. A horizontal, dashed, red line extends from the red dot to the
population cdf. At the intersection, a vertical, dashed red line extends
downward to the horizontal axis, where a second red dot, denoting the
associated gamma random variate is plotted.
This is not a particularly fast variate generation algorithm because it uses
the base R qgamma function to invert the values contained in u.
All of the elements of the u vector must be between 0 and 1.
Alternatively, u can be NULL in which case plot(s) of the
theoretical pdf and cdf are displayed according to plotting parameter values
(defaulting to display of both the pdf and cdf).
The user may specify a scale parameter or a rate parameter, but
not both.
The show parameter can be used as a shortcut way to denote plots to
display. The argument to show can be either:
a binary vector of length three, where the entries from left to right
correspond to showCDF, showPDF, and showECDF,
respectively. For each entry, a 1 indicates the plot should be
displayed, and a 0 indicates the plot should be suppressed.
an integer in [0,7] interpreted similar to Unix's chmod command. That
is, the integer's binary representation can be transformed into a
length-three vector discussed above (e.g., 6 corresponds to c(1,1,0)).
See examples.
Any valid value for show takes precedence over existing individual
values for showCDF, showPDF, and showECDF.
The minPlotQuantile and maxPlotQuantile arguments are present in
order to compress the plots horizontally. The random variates generated are
not impacted by these two arguments. Vertical, dotted, black lines are
plotted at the associated quantiles on the plots.
The plotDelay and maxPlotTime arguments can be used to slow down
the variate generation for classroom explanation.
In the plot associated with the pdf, the maximum plotting height is associated
with 125% of the maximum height of pdf. Any histogram cell that extends
above this limit will have three black dots appearing above it.