Maxwellian Probability Distribution Function
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Interpolation Function Method
We need to invert .
Here is a plot.
We see that the cumulative is quite close to 1 by the time .
The error (the "left over probability") in stopping at
this value of
is
If we sample
at a sequence of points, we can form ordered pairs of those points
with the
and
order reversed. From a list of such points we can then form an InterpolatingFunction
object through the use of Mathematica's Interpolation function.
One way to do this is to just use the Table
function as in
This method samples MaxwellCDF
at regularly separated points. Another way to do the sampling is
to make use of the adaptive algorithm that is used internally by
the Plot function.
The Graphics object
returned by the Plot
function above (and abbreviated as in
the output) contains within it the sampled data points that Plot's
internal algorithm obtained. We can see the structure of CDFPlot
by looking at its InputForm (and we wrap it in the function Short
so that not all of its contents are displayed).
The data that we are interested in is contained in the argument
of the Line graphics
primitive. This can be accessed by just taking the correct Part
of CDFPlot. Once
we have this we need to reverse the
and
order:
In the above function we extracted the data from the Graphics
function by directly accessing the needed Part
of the expression. Another way to do this is to use pattern
matching and transformation rules as follows.
then, .

We can now define an inverse function (using the data that was
generated by the Plot
command).
This interpolation function is not well defined outside of the
range of the sample points, given by the first argument of the InterpolatingFunction
(the second argument is suppressed in the output).

{{6.855672916839493*^21, 0.999999999920108}} 
Here is an example of a Maxwellian random number.
So we can extend the Random
function. We associate this extension with MaxwellPDF
rather than Random.
Now let's generate a list of 5000 Maxwellian random numbers.
To look at this data we can histogram it. First we load the standard
addon package, Statistics`DataManipulation`.
Now we can use the function BinCounts.
Here is the binned data.
Let's visualize the result in the form of a histogram along with
a plot of MaxwellPDF
for comparison.
