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Title: Trapezoidal And Triangular Distributions For Type B Evaluation Of Standard Uncertainty
Author: Raghu N Kacker, James F. Lawrence
Source: 2008 Measurement Science Conference
Year Published: 2008
Abstract: The Guide to the Expression of Uncertainty in Measurement (GUM), published by the International Organization for Standardization (ISO), recognizes Type B state-ofknowledge probability distributions specified by scientific judgment as valid means to quantify uncertainty. The ISO-GUM discusses symmetric probability distributions only. Sometimes an asymmetric distribution is needed. We describe a trapezoidal distribution which may be asymmetric depending on the settings of its parameters. We describe the probability density function (pdf), cumulative distribution function (cdf), inverse function of the cdf, moment generating function (mgf), moments about origin (zero), expected value, and variance of a trapezoidal distribution. We show that triangular and rectangular distributions are special cases of the trapezoidal distribution. Then we derive the moment generating functions, moments, expected values, and variances of various special cases of the trapezoidal distribution. Finally, we illustrate through a real life example how a Type B asymmetric trapezoidal distribution may be useful in quantifying a correction for bias (systematic error) in a result of measurement and in quantifying the standard uncertainty associated with the correction. An essentially identical version of this paper appeared in Metrologia, 44 (2007) 117-127. This is work of US federal government employees and hence not subject to copyright. Any references to commercial products in this document do not imply endorsement by the US National Institute of Standards and Technology.

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