The Failure Rate

Abstract

In our everyday life, we are frequently much more interested in the reliability of used goods than new ones. When someone buys a clock, the consumer has a warranty from the store ensuring that he can replace the clock with a new one or have it repaired if the clock fails within a certain period of time. Therefore, the ability of the clock to function without failure during this period of time does not trouble the user greatly. The question as to how long the clock will run after the period specified in the warranty is quite another matter. Failure of the clock then lies entirely on the shoulders of the purchaser. What the purchaser is interested in is that the probability that the clock will run without failure over a period of several years after the period specified in the warranty is sufficiently great. Let us look at this situation from a formal point of view. Let us denote by t_H a period of time during which the object in question functioned without failure and let us introduce the probability P{T_k > T | t_H} of failure-free operation during a period of time T after the object has functioned without failure for a time t_H. Thus, in the case of our present example, if the period of the warranty is a year, then t_H = 1. Wishing to determine the probability that the clock will continue to function for 20 years after the expiration of the warranty, we need to calculate the probability P{T_h > 20 | t_H = 1}. This probability is a conditional probability and, to evaluate it, we need to make use of the fact that an arbitrary distribution of the lifetime can be formally written

$$F\left( T \right)=1-\exp \left[ -\int\limits_{O}^{T}{\lambda \left( t \right)dt} \right]$$

(214)

where λ(t) denotes the function

$$\lambda \left( t \right)=\frac{f(t)}{1-F(t)}$$

(215)