Our main, novel, result is that a certain non-central chi-squared mixture of distributions itself scaled distribution. From this and link to known on representation for central distribution, numerous further results, both old new, ensue. These include results F, F Libby–Novick distributions. The main involves all with the same degrees freedom; it also extended case where mixing distribution has...

In the previous two Chapters, we proposed fraction nonconforming nonparametric control charts to monitor process location and process variability. It is also shown that performance of proposed control charts are superior to that of the Shewhart X and sign charts. If process is running in an in-control state for a long period, it will reach in steady-state mode. In order to characterize long-ter...

The cumulative distribution function of the non-central chi-square χn′2(λ) n degrees freedom possesses an integral representation. Here we rewrite this in terms a lower incomplete gamma applying two second mean-value theorems for definite integrals, which are Bonnet type and Okamura’s variant du Bois–Reymond theorem. Related results exposed concerning small argument cases (CDF) their asymptotic...

PURPOSE Evaluate factors contributing to ptosis after glaucoma surgery. METHODS Three-year retrospective chart review from January 1, 2012, to January 1, 2015, 157 eyes, 3 surgeons, at Boston Medical Center, to determine the incidence of ptosis and the effects of each variable contributing to ptosis at 3 months after surgery. Each variable was analyzed using the chi-square or independent samp...

Due to the imperfection of processes, quality some products may be unsatisfactory. Moreover, equipment failure can stop production for a while. Therefore, integrating triple concepts quality, maintenance, and inventory control has attracted attention. Triple are constituents pre-sale costs. Selling price warranty generally considered maintain market share maximize producer's profit. Quality mai...

In several recent papers log-concavity results and related inequalities for a variety of distributions were obtained. This work is supposed to derive a nearly complete list of corresponding properties concerning the cdf’s and some related functions for Beta as well as for central and non-central Chi-square and F distributions, where hitherto only partial results were available. To this end we i...

Computation of the non-central chi square probability density function is encountered in diverse fields of applied statistics and engineering. The distribution is commonly computed as a Poisson mixture of central chi square densities, where the terms of the sum are computed starting with the integer nearest the non-centrality parameter. However, for computation of the values in either tail regi...