In recent years the uniform distributions and their convolutions find such applications that are relevant to geodesy-more precisely-to the modern theory of errors: (i) The convolutions of uniform distributions have been applied to the error distribution arising from data processing; (ii) Within the framework of geodesy, outliers were assumed to be distributed with uniform distribution. Bearing in mind these new developments and integrating these isolated topics, in this paper new closed formulae for the probability density and distribution functions of the sum of independent uniform random variables with unequal supports are derived. A brief outline of the relevance of convolutions of uniform distributions to the theory of errors related to astronomy and geodesy is given in historical setting. Along with these, the origin of uniform distribution is discussed with special emphasis on the root of the theory of errors.
Summary We generalize a result of Lehmann on the comparison of location experiments with uniform distributions on intervals. We compare a location experiment consisting of uniform distributions on parallelepipeds with a location experiment consisting of uniform distributions on convex bodies. We show that the first experiment can only be more informative than the second one if the convex bodies in the second experiment are themselves parallelepipeds. Further we show that the length of the edges of these parallelepipeds must fulfill a condition similar to the condition on the length of the intervals in Lehmann’s result.