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A Note on the Convolution of the Uniform and Related Distributions and Their Use in Quality Control

Author

Listed:
  • Killmann Frank

    (Deutsche Bank AG, Taunusanlage 12, D-60262 Frankfurt, Germany)

  • von Collani Elart

    (Universität Würzburg, Sanderring 2, D-97070 Würzburg, Germany. collani@mathematik.uni-wuerzburg.de)

Abstract

Consider a product with quality characteristic X. Assume that the product is composed of n parts each with quality characteristic Xi, and and letX = X1 + . . . +XnAssume that there is a lower specification L and an upper specification U for X. Then the problem arises how to select suitable specifications for the quality characteristics Xi of the parts. If the type of distributions of the partial quality characteristics Xi are known, then for solving the specification problem, the distribution of X = X1 + . . . + Xn is needed.In case of measurable quality features, usually the normal model is assumed for the Xi and the problem of deriving the distribution of X is solved in the well-known way. However, generally the only feature of Xi which is known with certainty is its bounded support, which follows from technical conditions. In such a case the normal approximation which is based on an unbounded support of each Xi may lead to a distribution of X which does not reflect reality sufficiently well.In this paper, the uniform or a related distribution is assumed for the partial quality characteristics and an explicit expression for the distribution of the sum X is derived. The usefulness of the result is illustrated by an example taken from industrial practice.

Suggested Citation

  • Killmann Frank & von Collani Elart, 2001. "A Note on the Convolution of the Uniform and Related Distributions and Their Use in Quality Control," Stochastics and Quality Control, De Gruyter, vol. 16(1), pages 17-41, January.
    • Handle: RePEc:bpj:ecqcon:v:16:y:2001:i:1:p:17-41:n:3
    DOI: 10.1515/EQC.2001.17
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    Cited by:

    1. Ramezani, Mohsen & Geroliminis, Nikolas, 2012. "On the estimation of arterial route travel time distribution with Markov chains," Transportation Research Part B: Methodological, Elsevier, vol. 46(10), pages 1576-1590.
    2. Jorge Arevalillo, 2012. "Exploring the relation between the r* approximation and the Edgeworth expansion," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 75(8), pages 1009-1024, November.
    3. Pazour, Jennifer A. & Meller, Russell D., 2013. "The impact of batch retrievals on throughput performance of a carousel system serviced by a storage and retrieval machine," International Journal of Production Economics, Elsevier, vol. 142(2), pages 332-342.

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