# Speed is of the Essence

I enjoy talking with Phil Rogers , who teaches statistical analysis and managerial decision making, because he enjoys talking about his students.  His class at the University of Houston’s C. T. Bauer College of Business is filled with M.B.A. candidates who are already holding down managerial positions in their day jobs.

Phil worked for EXXON for many years, and he took on any number of operations research assignments for which he developed mathematical models.  So he likes to invite his students to bring to class the real-world questions facing them in their jobs, and they work together to find the best mathematical approaches to decision evaluation.

Not long ago, Phil told me about some  pretty sophisticated analyses of his students have produced of questions that I don’t usually associate with "managerial."   Patient deposits for organ transplants.  Allocation of turbines at a wind farm.  The United States’ bid to become the venue for 2022 World Cup soccer.

He thinks the key to helping people learn modeling techniques is speed: a short learning curve and tools that students can become comfortable with fast.  For instance, he likes Monte Carlo software that goes to work quickly for the students, without them having to understand how the software works.

This year he had a great opportunity to test this theory about speed.  Sinopec and CNPC, two big Chinese oil companies (big meaning numbers 15 and 25 or some such on the Fortune World 500) each sent a small group of senior managers to Houston to brush up on quantitative analysis.   Phil had three days to teach these execs how to build a mathematical model.  Three days was all the time they could afford to be away from their home offices.

How did it go?  His high-power students, he thought, did quite well.  Which is what he expected because, he says, if you can get up to speed quickly in a familiar, universal interface, they way you do in say, Microsoft Excel statistics, "you don’t have to be able to program in Fortran to build a model."

# Is the reciprocal of a positively skewed distribution also positively skewed?

@RISK (risk analysis using Monte Carlo, software for Excel) can be a simple yet effective tool to explore statistical concepts and properties of distributions.  For example, one interesting question is whether the reciprocal of a positively skewed distribution is positively or negatively skewed.

One’s first thought may be that such a reciprocal is negatively skewed. Of course, when reflecting on such an issue for the lognormal distribution it becomes clear that this is not the case. Since the lognormal distribution is the result of multiplying many independent random processes, the reciprocal of such a distribution is the result of multiplying the reciprocals of these individual distributions. Therefore the reciprocal of a lognormal distribution is itself lognormal, and hence always positively skewed.

Turning to triangular distributions the situation is not so clear. @RISK can be used to sample a distribution and calculate its reciprocal. The @RISK Statistics functions can be used to compute the moments of the input distributions (mean, stddev, skewness using e.g. RiskTheoSkewness etc) and the statistics for the reciprocals are available after the simulation (using RiskSkewness etc).

The left graph shows a range of triangular distributions which are either symmetric or positively skewed (as model inputs), and the right hand graph shows the (simulated) reciprocals. It is interesting to note that the reciprocal of the symmetric distribution is positively skewed, whereas as the parameters of the distribution are adjusted, the reciprocal may either be positively or negatively skewed, but with a general prevalence for positive skew.

These properties could of course be explored further through mathematical manipulations, many of which are not trivial. However, @RISK provides an easy and intuitive way to explore such issues for the best possible decision making under uncertainty.

Dr. Michael Rees
Director of Training and Consulting