The Reliability Function

The reliability function can be derived using the previous definition of the cumulative distribution function, Eqn. (3). The probability of an event happening by time t is given by:

(4)

In particular, this represents the probability of a unit failing by time t.

From this, we obtain the most commonly used function in reliability engineering, the reliability function, which represents the probability of success of a unit in undertaking a mission of a prescribed duration.

To mathematically show this, 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 (or γ) and t. So from Eqn.(4):

In this situation, there are only two states that can occur: success or failure. These two states are also mutually exclusive. Since reliability and unreliability are the probabilities of these two mutually exclusive states, the sum of these probabilities is always equal to unity. So then:

Conversely:

The Conditional Reliability Function

The reliability function discussed previously assumes that the unit is starting the mission with no accumulated time, i.e. brand new. Conditional reliability calculations allow one to calculate the probability of a unit successfully completing a mission of a particular duration given that it has already successfully completed a mission of a certain duration. In this respect, the conditional reliability function could be considered to be the "reliability of used equipment."

The conditional reliability function is given by the equation,

where:

t is the duration of the new mission, and
T is the duration of the successfully completed previous mission

In other words, the fact that the equipment has already successfully completed a mission, T, tells us that the product successfully traversed the failure rate path for the period from , and it will now be failing according to the failure rate from . (Note: If the underlying distribution has a constant failure rate, this has no effect.) This is used when analyzing warranty data.