Fundamentals of Actuarial Mathematics, 3rd Edition

17 The minimum failure time

17.1 Introduction

Suppose that T₁, T₂, …, T_m are failure times defined on the same sample space. In this chapter, we investigate the random variable

In other words, T is the time of the first failure to occur among the m different failure times that are possible. In particular, this will recast the material of Chapters 10 and 11 into a stochastic framework, and show that both joint-life theory and multiple-decrement theory are special cases of this general problem. In addition, it will provide more rigorous arguments for some of the results of those chapters that were obtained in an intuitive fashion. Finally it will deal with the important cases where the failure times need not be independent.

In the joint-life case, where we have a group of m lives numbered 1, 2, … , m, we can take T_i to be the future lifetime of the ith life, so that T is the failure time of the joint m-life status. In the multiple-decrement context, we can take T_i to be the time of failure from cause i in the associated single-decrement setting, so it is the failure time of the ith cause, assuming no other causes of failure are operating. Then, the random variable T is the time of failure in the multiple-decrement model. (In the machine analogy of Section 11.6, T_i would be the failure time of the ith part.)

17.2 Joint distributions

We wish to expand somewhat on our brief description ...