Rectangular distribution whose end points are not exactly known: curvilinear trapezoidal distribution

Abstract

Metrologists often represent the state of knowledge concerning a quantity about which scant specific information is available by a rectangular probability distribution. The end points are frequently specified by subjective judgment; therefore, they are inexactly known. If the states of knowledge about the end points may be represented by other (narrower) rectangular distributions, then the resulting probability distribution looks like a trapezoid whose sloping sides are curved. We refer to such a probability distribution as curvilinear trapezoid. Depending on the limits of rectangular distributions for the end points, the curvilinear trapezoidal distribution may be asymmetric. In a previous paper we had shown that if the mid-point of a rectangular distribution is known and the state of knowledge about the half-width may be represented by a rectangular distribution then the resulting distribution is symmetric curvilinear trapezoid. In this paper, we describe the probability density function of a curvilinear trapezoidal distribution which arises from inexactly known end points. Then we give compact analytic expressions for all moments including the expected value and the variance. Next we discuss how random numbers from such a distribution may be generated. We compare the curvilinear trapezoid which arises from inexactly known end points with the corresponding trapezoid whose sloping sides are straight. We also compare the curvilinear trapezoid which arises from inexactly known end points with the curvilinear trapezoid which arises when the mid-point is known and the state of knowledge about the half-width may be represented by a rectangular distribution. The results presented in this paper are useful in evaluating uncertainty according to the Guide to the Expression of Uncertainty in Measurement (GUM) as well as Supplement 1 to the GUM (GUM-S1).