The Reliability Function

The reliability function can be derived using the previous definition of the cumulative density function, Eqn. (3). Note that the probability of an event occurring by time t based on a continuous distribution given by f(x), or henceforth f(t) since our random variable of interest in life data analysis is time, or t, is given by:

(4)

One could equate this event to the probability of a unit failing by time t.

From this fact, the most commonly used function in reliability engineering, the reliability function, can then be obtained. The reliability function enables the determination of the probability of success of a unit, in undertaking a mission of a prescribed duration.

To show this mathematically, we first define the unreliability function, Q(t), which is the probability of failure or the probability that our time-to-failure is in the region of 0 and t. So from Eqn. (4),

Reliability and unreliability are success and failure probabilities, are the only two events being considered and are mutually exclusive, hence the sum of these probabilities is equal to unity. So then:

Conversely: