Spare-parts problem [BARL 1981]. A system requires k components of a certaintype to function properly. All components are assumed to have a constant fail-ure rate , and their lifetimes are statistically independent. During a mission,component j is put into operation for tj time units, and a component can failonly while in operation. Determine the number of spares needed (in a commonsupply of spares) in order to achieve a probability greater than for the missionto succeed. As an example, let k = 3, t1 = 1300, t2 = 1500, t3 = 1200, = 0.002,and = 0.90, and determine the number of needed spares n. Now consider thealternative strategy of keeping a separate supply of spares for each of the threecomponent types. Find the number of spares n1, n2, and n3 required to providean assurance of more than 90% that no shortage for any component type willoccur. Show that the former strategy is the better one.

STAT 2000 Chapter 1: Learning from Data Defining Statistics: Statistics is a way of reasoning along with a collection of tools and methods, designed to help us understand the world Statistics is the scientific discipline, which consists of: o Formulating a research question o The collection of data o Describing data o Drawing conclusions or generalizations from data Our ability to answer questions and draw conclusions from data depends largely on our ability to understand variation . The key to learning from data is understanding the variation that is all around us. Definition: Statistics, as a field of study, is the science of learning from data. This definition raises two qu