[Maxima] Ideas welcome
Richard Hennessy
rich.hennessy at verizon.net
Wed Aug 5 15:17:46 CDT 2009
Thanks you very much for this. I didn't think of this application myself. I am glad you shared this. One thought I had was that a
Gaussian would always have a nonzero probability of being negative. I am sure you thought of that. Anyway c1*abs(x)^c2*exp(-a*x^2)
for some a>0, c1>0 and c2>0 might work better.
Rich
----- Original Message -----
From: "Jack Schmidt" <jack.schmidt at sbcglobal.net>
To: <maxima at math.utexas.edu>
Sent: Wednesday, August 05, 2009 1:17 AM
Subject: Re: [Maxima] Ideas welcome
dlakelan at street-artists.org wrote:
> IMHO The best method for making money while using maxima and other free
> software programs is to do it as a consultant. The key is to find some
> people who have specific real-world type problems to be solved, but do
> not have the sophistication to write the software to solve them.
>
Allow me to offer a pw.mac application illustrating Dan's point. (I was
going to do this myself but never got around to it.)
In real-world statistical analysis one often has many random variables
whose probability distributions can only be defined piecewise. For
example, the diameter of a ball bearing may have a natural Gaussian
distribution, but manufacturers reject bearings that are too large or
too small, so the ball bearing users see a truncated Gaussian -- a
piecewise distribution. Suppose a user plans to install three bearings
in a linear channel whose length is just a little greater than three
nominal bearing diameters. What is the probability that the three
bearings will fit? Most people answer this by ignoring the truncations
caused by the manufacturer's rejection criterion (thus being overly
conservative), and convolving the three pure Gaussians. Or they model
only the worst case -- all bearing diameters at the upper limit -- which
is OK when over-designing is acceptable. Or they address the problem
with a Monte Carlo simulation of 10^5 or 10^6 assemblies. But what if
you're building a million assemblies and you can only accept, say, 5
failures? (This is in the "Six Sigma" range.) The pure-Gaussian model
may predict failure rates several times the actual rate, and the Monte
Carlo technique will have to simulate maybe 10^7 or 10^8 assemblies,
which is possible in this case only because the calculation involved in
combining the three random variables -- summing the three randomly
generated diameters -- is simple. And this is just three random
variables. I sometimes deal with 30 or 40.
Instead, we could model the problem symbolically -- the symbolic
convolution of three piecewise-defined functions, yielding another
piecewise function (with many more pieces) to get a final probability
distribution function. Pretty easy, once you have pw.mac. Calculation
of the failure rate is then just a matter of (piecewise) integrating the
resulting piecewise function from the channel length to positive infinity.
This type of problem occurs with surprising yet unrecognized frequency
in the engineering world (and virtually never, I would guess, in
academia or science).
--
Jack
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