For, the density function of the time to failure, f(t), and the reliability function, R(t), the hazard rate function for any time, t, can be defined as
h(t) = f(t) / R(t)
Where, f(t) is the probability density function (PDF) representing a failure distribution and R(t) is the survival function.
In words: the probability that if you survive to t, you will succumb to the event in the next instant.
Average failure rate is the fraction of the number of units that fail during an interval by the number of units alive at the beginning of the interval.
In the limit of smaller time intervals, the average failure rate measures the rate of failure in the next instant on time for those units (conditioned on) surviving to time t, known as instantaneous failure rate,
Hazard vs. Density
When you are born, you have a certain probability of dying at any age; that’s the probability density.
Example: a woman born today has, say, a 1% chance of dying at 80 years.
However, as you survive for awhile, your probabilities keep changing
Example, a woman who is 79 today has, say, a 5% chance of dying at 80 years.
Hazard rates are applied to non repairable systems.