Derivation of isosceles trapezoidal distributions

Abstract

The Guide to the Expression of Uncertainty in Measurement (GUM) (ISO 1995, Geneva: International Organization for Standardization) notes that an isosceles trapezoidal distribution often expresses the state of knowledge probabilities better than a rectangular distribution. The GUM lends further support to an isosceles trapezoidal distribution by interpreting it as a rectangular distribution whose inexactly known half-width may be represented by a (narrower) rectangular distribution. This interpretation is not exactly correct. It turns out that if the inexact knowledge about the half-width is represented by a rectangular distribution, then the resulting distribution is a variation of the isosceles trapezoid whose sloping sides are curved, an isocurvilinear trapezoidal distribution. The isosceles trapezoid can be regarded as an approximation to the isocurvilinear trapezoid. Therefore the GUM's interpretation is approximately correct. Question: does a probability distribution exist for the inexactly known half-width for which the resulting distribution is the isosceles trapezoid recommended in the GUM? We show that the isosceles trapezoidal distribution results when the inexact knowledge about the half-width is represented by a truncated right triangular distribution. The truncated right triangular distribution looks like a modification of the rectangular distribution whose top is sloping. The required truncated right triangular distribution has the same midpoint and the same second moment about the midpoint as the rectangular distribution on the same interval. It is often difficult for a metrologist to express the state of inexact knowledge about the half-width in terms of a well-defined probability distribution function. However, a metrologist can be expected to specify approximate limits for the inexactly known half-width. At least two probability distributions exist for the half-width for which the resulting distribution is isosceles trapezoid exactly or approximately. Therefore the isosceles trapezoidal distribution recommended in the GUM may be used as an approximation when only approximate limits for the half-width can be specified.