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# Rectangular Distribution Whose End-points Are Not Exactly Known: Curvilinear Trapezoidal Distribution

Published

### Author(s)

Raghu N. Kacker, James F. Lawrence

### Abstract

The state of knowledge concerning a quantity about which scant specific information is available is often represented by a rectangular probability distribution on some interval (Z1, Z2) specified by scientific judgment. Often, the end-points Z1 and Z2 are not exactly known. If the state of knowledge about the left end-point Z1 can be represented by a rectangular distribution on the interval (a, c) and the state of knowledge about the right end-point Z2 can be represented by a rectangular distribution on the interval (d, b), where a ≤ c ≤ d ≤ b, then the resulting probability distribution looks like a trapezoid whose sloping sides are curved. We can refer to such a probability distribution as curvilinear trapezoidal distribution. Depending on the values of a, c, d, and b, the curvilinear trapezoidal distribution may be asymmetric. We describe the probability density function (pdf) and the moments of a curvilinear trapezoidal distribution which arises from inexactly known end-points of a rectangular distribution. In particular, we give compact algebraic expressions for the expected value and the variance. Then we discuss how random numbers from such a distribution may be generated. We compare the curvilinear trapezoidal distribution which arises from inexact end-points with the corresponding trapezoidal distribution whose sloping sides are straight. We also compare the curvilinear trapezoidal distribution which arises from inexactly known end-points with the curvilinear trapezoidal distribution which arises when the mid-point of a rectangular distribution is known (fixed), the half-width is not exactly known, and the state of knowledge about the half-width may be represented by a rectangular distribution.
Citation
Metrologia
Volume
47

### Keywords

Rectangular distribution, trapezoidal distribution, uncertainty in measurement

## Citation

Kacker, R. and Lawrence, J. (2010), Rectangular Distribution Whose End-points Are Not Exactly Known: Curvilinear Trapezoidal Distribution, Metrologia, [online], https://tsapps.nist.gov/publication/get_pdf.cfm?pub_id=903235 (Accessed June 25, 2024)