Modern Mathematics for the Engineer: Second Series by Beckenbach Edwin F.; Hestenes Magnus R.;

Author:Beckenbach, Edwin F.; Hestenes, Magnus R.;
Language: eng
Format: epub
Publisher: Dover Publications
Published: 2013-03-03T16:00:00+00:00

and is the unique solution of the equation

The fact that the solution is unique simplifies significantly the computation of . Also, in this case any statistical estimate of ϕ can be translated quickly into a reliable statistical estimate of the critical number .

The class of densities that are Pólya frequency functions includes a large number of the standard unimodal densities arising in statistical applications. If ϕ(ξ) is a density of a gamma distribution of integral order, a truncated normal distribution, or a Kolmogoroff-Smirnoff distribution, it is a Pólya frequency function. In the case of discrete distributions, the Poisson and the Pascal family (negative binomial) qualify as discrete Pólya frequency functions. For a detailed discussion of Pólya frequency functions, the reader is referred to Refs. 11, 15, and 16.

As another example, we discuss now the case in which c(z) is strictly convex and increasing, with h and p also convex functions.

Let y*(x) denote the unique root of c′(y − x) + G′(y) = 0 (whenever it exists), where

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