| Summary: | 博士 === 淡江大學 === 管理科學研究所 === 69 === This thesis contains five main Chapters. In Chapter 1, we characterize the general distributions by a sequence of linear combinations of odd order moments of the order statistics; and by an odd order moment and a sequence of linear combinations of even order moments of the order statistics. In Chapter 2, we extend the well-known Muntz-Szasz Theorem and three complete sequences (the main tools in Chapter 1) by considering the non- negative functions which are absolutely continuous and have derivatives nonzero almost everywhere on an interval [a,b]. Then we answer a moment problem posed by Hwang, Pan and Wu, and extend some results of Huang and Hwang. In Chapter 3, we characterize two parameter distributions by three different ways : the inverse function of the distribution, two linear combinations of the expected values of the order statistics, and the bounds of the expected values of the order statistics. We also answer two problems about the uniform, the exponential and the normal distri- butions, which were posed by Hwang and Jenq. In Chapter 4, we apply the theoretical results in Chapter 3 to construct a new pair of variables control charts, X1.2-X2.2 charts, which in many cases is better than the well-known X-R charts. In Chapter 5, we utilize the technique of the inverse function of the distribution to trans- form one kind of chance-constrained programming problems into deterministic-constrained programming problems. Then we provide a set of optimality conditions for mathematical programming problems with linear constraints. Finally, we provide two counterexamples, one for a theorem of Singh in mathematical pro- gramming, and the other for a theorem of Zionts in integer pro- gramming, which can be modified by adding a suitable condition.
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