Failure rate function

The failure rate function expresses the conditional probability that an item that has been functioning since it was put into function fails in a small time interval. The notation z(t) is used to express the failure rate function. The failure rate function may be expressed by the probability density function and the survivor function:

$ \color{blue}{z(t) = \lim_{{\Delta }t\to 0}\frac{{\scriptsize \mbox{Pr}}(t \le T \lt t+ \Delta t | t > T)}{\Delta t} = \frac{f(t)}{R(t)} = \frac{f(t)}{1-F(t)}} $

Or rewritten as:

$ \color{blue}{z(t) \Delta t \approx \mbox{Pr}(t \le T \lt t+ \Delta t | t > T)} $

which means that the failure rate function multiplied by the length of a small time interval, $[t , t+ \Delta t]$, may be approximated with the probability that the item fails in the interval given that it is still functioning at the begining of the intervl, i.e., at time t.

The failure rate function may also be interpreted as the conditional probability of a failure in a small time interval given survival up to time t, and then divided by the length of the interval.

The figure below shows a typical example of a failure rate function where z(t) first is decreasing (burn-in problems), then rather constant before it starts increasing due to wear-out:

The term bathtub curve is often used due to this characteristic behaviour.