Abstract

The distribution densities of the laws of distribution of random operating times known in the theory of reliability, for example, exponential, Weibull–Gnedenko, Erlang, normal, Maxwell, and many others are not more than unimodal and not more than two-parameter. This paper discusses formulation of problems of determining the share of each distribution function in a mixture (distribution functions are given) for which a random variable specified by this mixture of distributions has the least variance under a certain expectation or the largest expectation under a certain variance. The problems are formulated as the well-known Markowitz problems on selecting a block of securities under the assumption that the mean represents the return and the variance represents the risk. Solutions to the problem of minimizing the variance for mixtures of two and three distributions for a certain expectation are presented.

中文翻译：

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技术系统要素失效运行时间分布函数组合的优化问题

摘要

可靠性理论中已知的随机操作时间分布规律的分布密度，例如指数、Weibull-Gnedenko、Erlang、正态、麦克斯韦等，不超过单峰，不超过二参数。本文讨论了确定混合物中每个分布函数的份额（给出了分布函数）的问题的公式化，其中由该分布混合物指定的随机变量在某个期望下具有最小方差或在某个方差下具有最大期望. 这些问题被表述为众所周知的 Markowitz 问题，即在均值代表收益而方差代表风险的假设下选择一组证券。