Median Inactivity Time Function and its Reliability Properties

Let the non-negative random variable X denote the life time of a unit, then the random variable X (t) = t – X for X ≤ t for a fixed t ∈ {x : FX (x) > 0}, is known as inactivity time or reversed residual life time. In this paper, we define a new non-parametric class of life distributions based on the median of the random variable X (t) and study its reliability properties. Some new results concerning the proposed class are given including some closure properties and characterizations. We also introduce and study a new stochastic ordering based on the median of X (t) and find its relationship with other well-known order relations. Finally, we provide the median inactivity time function of some well-known life time distributions.