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

Abstract

Consider a product with quality characteristic X. Assume that the product is composed of n parts each with quality characteristic X_i, and and let

X = X₁ + . . . +X_n

Assume 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 X_i of the parts. If the type of distributions of the partial quality characteristics X_i are known, then for solving the specification problem, the distribution of X = X₁ + . . . + X_n is needed.

In case of measurable quality features, usually the normal model is assumed for the X_i and the problem of deriving the distribution of X is solved in the well-known way. However, generally the only feature of X_i 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 X_i 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.