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Compensating for “Lack of Tails” on Triangular Distributions

Risk Track

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The use of triangular distributions for capturing cost risk has been popular because they have exceptional attributes for the collection of expert information on the uncertainties of cost. However triangles have one major drawback, the absolute limitation of risk at the extremes, i.e. the Low and High values have zero percent probability of exceedence. Conversely the normal or Gaussian distribution has infinite “tails” but information collection, although requiring only two parameters (mean and sigma), is often difficult to elicit from experts.

This paper uses two distributions (Triangular and Normal) as representative of two classes of probability distributions to show a solution that offers minimum impact on the underlying cost risk while expanding the probability outcomes beyond that of High and Low values. In addition the methodology for adjusting triangles for the lack of infinite “tails” was extended to lognormal like distributions.

Aside from providing compensation mechanisms for the limitations of the finite Low and High triangular values, the paper also offers uses for a newly hypothesized triangle called the “lognormal triangle‐fitted” distribution. One example of its use is the application of the ratio of the High over Low value as an easy check to see how close a triangular risk input is to an underlying lognormal distribution.


Melvin Broder
The Aerospace Corporation
Melvin A. Broder is a Senior Project Leader at The Aerospace Corporation working in the System Planning and Engineering Group. In that capacity he has developed cost models for the Concept Design Center, building and expanding tools for the cost seat and devising new processes. Prior to joining Aerospace he worked in cost estimating at Boeing’s Satellite Systems where he was responsible for front end of the business cost tools and models for Boeing’s commercial product line. Other work included support of the BSS Design Center’s Integrated Engineering Laboratory with the creation of several automated cost tools, which uses a spacecraft’s system-level set of parameters to generate costs. Mr. Broder has also been a Project Manager for cost tools and processes in the System Engineering Laboratory at Raytheon Systems Company. His responsibilities include the creation and maintenance of tools to support the Sensors and Electronic Systems in design to cost (DTC) and cost-as-an-independent-variable (CAIV) activities. Prior to working the aerospace industry he was an Instructor of Economics at La Verne College, teaching both upper and lower division course work. He earned an M.S. in Economics, specializing in econometrics from the University of Southern California.