The hazard function is a measure of the tendency to fail; the greater the value of the hazard function, the greater the probability of impending failure. Technically, the hazard function is the probability of failure in the very small time interval, x0 to x0 + dx given survival until x0. The hazard function is also known as the instantaneous failure rate. Mathematically, the hazard function is defined as
Using the expression above, and the two expressions below, if either the hazard function, reliability function, or probability density function is known, the remaining two functions can be derived.
If the hazard function is increasing, failures are caused by wearout. If the hazard function is decreasing infant mortality failures are occurring. Some causes of infant mortality failures are:
Example: Given the hazard function, h(x) = 2x, derive the reliability function and the probability density function.
Solution: The reliability function is
The probability density function is