Hazard Probability Function

reading prerequisites: Moment-Generating Distribution Functions

Hazard Probability Function is used to express the “risk” that a component fails at time t (note: a component fails fluctuates based on time)

Hazard Function (∆t dependent)

h(t, ∆t) = P(t ≤ T ≤ t + ∆t | T ≥ t)

Probability of random variable T being within t and t + ∆t, given that random variable T is greater than or equal to t

To make P(t ≤ T ≤ t + ∆t | T ≥ t) independent of ∆t, we consider the limit ∆t → 0. Then, however, the probability would always tend to 0. To avoid this, we divide it by ∆t, thus obtaining a quantity that is similar in spirit to the probability-density-function. This gives the definition:

Hazard Function (∆t independent)

h(t) = lim P(t ≤ T ≤ t + ∆t | T ≥ t) / ∆t
      ∆t→0

h(t) = pdf(t) / ∫_-t^∞pdf(t)dt

where:

• pdf = probability density function

since a typical probability-density-function sooner or later starts decreasing with time, this effect tends to diminish the hazard as t increases. On the other hand, the denominator decreases as t increases, which will increase the hazard. These 2 opposing effects can balance each other in different ways.

Hazard Function (special case)

a special case when they precisely balance each other is the exponential distribution, where the probability-density-function is of the form:

pdf(t) = ke⁻ᵏᵗ

In this case the hazard function obtained is a constant:

h(t) = k

Hazard Function (Common Properties)

In many cases the hazard function has a bathtub shape that consists of 3 parts:

• initial “burn-in” period, when the hazard is relatively large, due to potential manufacturing defects that result in early failure
• steady part with approximately constant hazard function
• aging with increasing hazard