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Trapezoidal and Triangular Distributions for Type B Evaluation of Standard Uncertainty

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Trapezoidal and Triangular Distributions for Type B Evaluation of Standard Uncertainty IOP PUBLISHING METROLOGIA Metrologia 44 (2007) 117–127 doi:10.1088/0026-1394/44/2/003 Trapezoidal and triangular distributions for Type B evaluation of standard uncertainty Raghu N Kacker1 and James F Lawrence1,2 1 Mathematical and Computational Science Division, National Institute of Standards and Technology, Gaithersburg, MD 20899, USA 2 Department of Mathematical Sciences, George Mason University, Fairfax, VA 22030, USA E-mail: [email protected] Received 6 September 2006 Published 26 February 2007 Online at stacks.iop.org/Met/44/117 Abstract The Guide to the Expression of Uncertainty in Measurement (GUM), published by the International Organization for Standardization (ISO), recognizes Type B state-of-knowledge 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, cumulative distribution function (cdf), inverse function of the cdf, moment generating function, 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. 1. Introduction trapezoidal distribution, which may be asymmetric depending on the settings of its parameters. We give explicit and simple The International Organization for Standardization (ISO) expressions for the moment generating function (mgf) and Guide to the Expression of Uncertainty in Measurement the moments of a trapezoidal distribution. In particular, (GUM) recognizes Type B (non-statistical) evaluations. A we give expressions for the expected value and variance Type B estimate and standard uncertainty for an input of a trapezoidal distribution. These expressions are new quantity are the expected value and standard deviation contributions to the literature. We also discuss how random of a state-of-knowledge probability distribution specified numbers from a trapezoidal distribution may be numerically by scientific judgment based on all available information, generated. In section 3, we discuss various special cases [1, section 4.3]. The ISO-GUM [1] discusses the of the trapezoidal distribution, which include triangular and following probability distributions for a Type B input rectangular distributions. Then we derive the mgfs, moments, variable of the measurement equation: normal (Gaussian) distribution, rectangular (uniform) distribution, isosceles expected values and variances of various special cases of the triangular distribution and isosceles trapezoidal distribution. trapezoidal distribution. These distributions are symmetric about their expected values A particular use of a Type B probability distribution and they are useful in many applications. However, in is to quantify a correction for bias (systematic error) in some applications an asymmetric distribution is needed [1, a result of measurement. The exact bias is unknowable; sections 4.3.8, F.2.4.4 and G.5.3]. In section 2, we describe a therefore, a correction for bias carries uncertainty. According 0026-1394/07/020117+11$30.00 © 2007 BIPM and IOP Publishing Ltd Printed in the UK 117 R N Kacker andJFLawrence f (x) Trapezoid (a, c, d, b),is f(x)= 0ifx a, (2.1) h (x − a) f(x)= h if a x c, (2.2) (c − a) = X f(x) h if c x d, (2.3) a c d b (b − x) f(x)= h if d x b, (2.4) rs t (b − d) f(x)= 0ifb x, (2.5) Figure 1. Probability density function of Trapezoid (a, c, d, b). where to the ISO-GUM, a result of measurement that is subject to 2 2 h = = . (2.6) non-negligible bias should be corrected3 and the uncertainty [(b − a) + (d − c)] (r +2s + t) associated with the correction should be quantified and The cumulative distribution function (cdf) F(x), defined as included in the combined standard uncertainty associated F(x) = Pr(X x), is obtained by integrating the pdf of with the corrected result [1, section 3.2]. This is a signal X defined above in (2.1)–(2.6), within the limits from −∞ contribution of the ISO-GUM because it establishes, for the to x. Thus, the cdf F(x) of the trapezoidal distribution, first time, an approach to account for the uncertainty from Trapezoid (a, c, d, b),is unknown bias. A correction for bias and its associated standard uncertainty are generally the expected value and F(x) = 0ifx a, (2.7) standard deviation of a Type B state-of-knowledge probability 4 h(x − a)2 distribution for an input variable which represents a correction F(x) = if a x c, (2.8) (or a correction factor) in the measurement equation. In 2(c − a) section 4, we review the ISO-GUM’s approach to quantify and h F(x) = (c − a) + h(x − c) if c x d, (2.9) incorporate a correction for bias and its associated standard 2 uncertainty. In section 5, we illustrate through a real life h(b − x)2 example how a Type B asymmetric trapezoidal distribution F(x) = 1 − if d x b. (2.10) (b − d) may be useful in quantifying a correction for bias and its 2 associated standard uncertainty. A summary appears in and section 6. F(x) = 1ifb x. (2.11) In particular, = 2. Trapezoidal distribution F(a) 0, (2.12) h F(c) = (c − a), (2.13) A random variable X has a trapezoidal distribution if its 2 probability density function (pdf) f(x) has the shape of a h [(d − a) + (d − c)] F(d) = 1 − (b − d) = (2.14) trapezoid shown in figure 1. A trapezoidal distribution may 2 [(b − a) + (d − c)] be defined by four parameters, a, c, d and b, where a and b are and the end points and the points c and d identify the rectangular F(b) = 1. (2.15) part of the trapezoid. We refer to such a distribution as − Trapezoid (a, c, d, b), where a c, c d and d b.We The inverse function x = F 1(y) of the cdf F(x)is determined use the symbols r, s and t for the widths of the line segments by setting y = F(x), for 0 y 1, and solving equations − between a, c, d and b, where r = (c − a), s = (d − c) and (2.8)–(2.10) for x. Thus the inverse function x = F 1(y) of t = (b − d). Clearly r + s + t = (b − a) is the width of the the cdf F(x)is trapezoid. 2(c − a) √ The area of the trapezoid shown in figure 1 is h(c −a)/2+ F −1(y) = a + × y h h(d − c) + h(b − d)/2. For a trapezoid to represent a pdf (2.16) its area should be one. By equating the area h(c − a)/2+ h − − if 0 y (c − a), h(d c) + h(b d)/2 to one and solving for h we get the 2 height h = 2/[(b − a) + (d − c)] = 2/(r +2s + t). Then (a + c) y algebraic equations of the straight lines in figure 1 give the pdf. F −1(y) = + 2 h Thus, the pdf f(x)of the trapezoidal probability distribution, (2.17) h h 3 The exception is when one is highly certain (very sure) that the bias is if (c − a) y 1 − (b − d), negligible. This corresponds to the situation where both the expected value 2 2 and the standard deviation of a state-of-knowledge probability distribution for 2(b − d) the correction variable are close to zero. F −1(y) = b − × 1 − y 4 There is no direct correspondence between the classification of uncertainties h as Type A or Type B evaluations and the classification of the uncertainties (2.18) h as arising from random effects or those associated with the corrections for if 1 − (b − d) y 1. systematic effects [2, section D2]. 2 118 Metrologia, 44 (2007) 117–127 Trapezoidal and triangular distributions for Type B evaluation of standard uncertainty In particular, V(X)= E(X − E(X))2, which is the second central moment −1 F (0) = a, (2.19) (moment about mean). The expected value E(X) is obtained h by substituting k = 1in(2.26). Thus we have F −1 (c − a) = c, (2.20) 2 h b3 − d3 c3 − a3 h E(X) = − . (2.27) F −1 1 − (b − d) = d (2.21) 6 b − d c − a 2 and The expected value (2.27) can alternatively be expressed as F −1(1) = b, (2.22) (b2 − a2) + (d2 − c2) − ac + bd as one would expect in view of the equations from (2.12)to E(X) = . (2.28) 3[(b − a) + (d − c)] (2.15). The second moment, E(X2), is obtained by substituting k = 2 2.1. Moment generating function and moments of Trapezoid in (2.26). Thus we have (a,c,d,b) h b4 − d4 c4 − a4 The mgf of the pdf of a random variable X, denoted by M(t), E(X2) = − . (2.29) is defined as M(t) = E(etX), provided that this expected value 12 b − d c − a exists in some neighbourhood of zero5. As the name suggests, an mgf can be used to generate the moments about origin (zero), The second moment (2.29) can alternatively be expressed as E(Xk), for k = 1, 2,...
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