A Stochastic Study of the Life Table and Its Applications: I. Probability Distributions of the Biometric Functions

The life table is one of the oldest, most useful, and best-known topics in the field of statistics. It has many applications in various areas of research where birth, death, and illness may take place. The earliest life tables date as far back as the seventeenth century; Halley's famous table for the City of Breslau, published in the year 1693 9, already contained most of the columns in use today. The subject matter, however, is by no means limited to human beings. Zoologists, biologists, physicists, manufacturers, and investigators in other fields have found the life table a valuable means of presenting their data. In spite of its popularity in many research areas, the life table as a subject has yet to be systematically explored from a statistical point of view. There are two forms of the life table in general use: the cohort (or generation) life table and the current life table. In its strictest form a cohort life table records the actual mortality experience of a given group of individuals over a period of time extending from birth until the death of the last member of the group. A current life table, on the other hand, considers the mortality experience of an entire population at one point in time. The purpose of this investigation is to present a stochastic view of the subject, taking random variation into consideration and treating all the biometric functions as random variables. The results of our study will be given in a series of papers. In the first paper probability distributions of the main biometric functions are presented and formulas are derived for the corresponding mathematical expectations, variances, and covariances. Some of the findings are by no means original, but they are included for the sake of completeness.