On the discrete mean past lifetime

Reversed variance residual life function and its properties in discrete lifetime models

Purpose

In reliability studies, interests in discrete failure data came relatively late in comparison to its continuous analogue. Also, discrete failure data arise in several common situations. So, in this paper the authors try to study some reliability concepts such as reversed variance and reversed mean residual life functions based on discrete lifetime random variable.

Design/methodology/approach

Supposed T be a non‐negative discrete random variable, then based on reversed residual random variable Tk*=(k−T|T≤k), some useful and applicable relations and bounds are achieved.

Findings

In this paper, the authors study the reversed variance residual life in discrete lifetime distributions, the results of which are not similar to the continuous case. Its relationship with reversed mean residual life and reversed residual coefficient of variation are obtained. Also, its monotonicity and the associated ageing classes of distributions are discussed. Some characterization results of the class of increasing reversed variance residual life, which is denoted by IRVR, are presented and the upper bound for reversed variance residual life under some conditions is obtained.

Practical implications

There are many situations where a continuous time is inappropriate for describing the lifetime of devices and other systems. For example, the lifetime of many devices in industry, such as switches and mechanical tools, depends essentially on the number of times that they are turned on and off or the number of shocks they receive. In such cases, the time to failure is often more appropriately represented by the number of times they are used before they fail, which is a discrete random variable.

Originality/value

All the results based on discrete reversed residual lifetime, such as the relationships among reversed mean, variance and coefficient of variation residual lifetime and also their monotonicity ageing classes, are new.