Statistics for Maintenance Modelling: Continuous Lifetime Distributions 1

# Determining the Probability Density Function From the Hazard Function¤

Often it is easier to specify/approximate the hazard function than the probability density function, e.g. the hazard function might be specified to follow a piecewise linear bath tub curve, shown below. The probability density function is then determined from inversion of the hazard function.

Note that:

- if R(t) 1, then h(t) f(t)
- the first step in finding the probability density function from the hazard function is to see that the hazard function is the negative of the derivative of the reliability function so that integrating minus the hazard function gives the log of the reliability function:
  , then
- the cummulative hazard function is H(t) = h(u) du, and it is readily seen that R(t) = e-H (t)