Product Warranty Period: A Markovian Approach to Estimation and Analysis of Repair and Replacement Costs--A Comment.
本文是Franklin Lowenthal对Balachandran等人1981年论文的评论，指出其中存在数学错误：方程(3)错误地表达了时间独立性条件，实际是概率论中的事件独立性概念，可能导致读者得出错误结论。
Abstract The article presents information on comments made by Franklin Lowenthal on the article "Product Warranty Period: A Markovian Approach to Estimation and Analysis of Repair and Replacement Costs," by K.R. Balachandran, R.A. Maschmeyer and J.L. Livingston. Lowenthal has said that the article by Balachandran, Maschmeyer and Livingston [1981] contains a mathematical error. The conclusion that the exponential distribution is the only distribution that leads to a failure rate constant during the warranty period is certainly correct. However, the mathematical formulation of the postulated condition of time independence as expressed in Equation (3) on page 117 in that paper is incorrect. The incorrect equation expresses a quite different but exceedingly fundamental concept of probability theory: independence of events. In fact, the word "independence," which generally refers to independence of events, is used by the authors immediately prior to their incorrect equation. A reader who attempted to combine the incorrect Equation (3) and correct Equation (1) would be led to an incorrect conclusion.