Lifetimes

Consider a light fitting in which the light bulb is replaced whenever it fails. If the failure rate, \(\lambda\), remains constant over time and the risk of failure at any time does not depend on what happened previously, failures would be a homogeneous Poisson process.

We will now concentrate on a single light bulb. Its lifetime is the time until the first event in this Poisson process and would have an \(\ExponDistn(\lambda)\) distribution.

Lifetimes may be modelled as the time until the first event of a Poisson process.

This model does not however allow for the possibility of items improving or deteriorating over their lifetime.

Exponential hazard function

The survivor function for the exponential distribution is

\[ S(x) \;\; = \; \; 1 - F(x) \;\;=\;\; e^{-\lambda x}\]

and its hazard function is

\[ h(x) \;\; = \; \; \frac {\lambda e^{-\lambda x}}{e^{-\lambda x}} \;\;=\;\; \lambda\]

This constant hazard function corresponds to the "memoryless" property of Poisson processes — the chance of failing does not depend on what happened before and, in particular, how long the item has already survived.

We will consider other distributions for lifetimes in which the hazard function changes as the items age later.