Hazard Rate

Hazard Rate

What it is

The hazard rate measures the instantaneous likelihood that a non-repairable item will fail at a specific age, given that it has survived up to that age. It is a core concept in survival analysis (also called reliability analysis, duration analysis, or event-history analysis depending on the field) and is widely used in engineering, medicine, insurance, and finance to assess risk and plan maintenance or interventions.

Key points

• The hazard rate indicates the conditional probability of failure at time t, given survival to t.
• It is often called the failure rate.
• The hazard rate cannot be negative.
• Many real-world systems show a characteristic “bathtub” shape in their hazard-rate curve: early failures, a stable useful-life period, then increasing failures as items wear out.

How it’s calculated

The hazard rate h(t) at time t is defined as:
h(t) = f(t) / R(t)

Where:
* f(t) is the probability density function (PDF) for failure at time t — the probability that failure occurs in a small interval around t.
* R(t) is the survival function — the probability the item survives past time t.

This ratio gives the instantaneous failure rate conditional on survival to time t.

Interpreting the curve

Common hazard-rate patterns:
* Decreasing early hazard (“infant mortality”): Early defects produce higher initial failure rates that decline once defective units are removed.
* Constant hazard (useful life): Failures occur randomly at a roughly steady rate.
* Increasing hazard (wear-out): Aging or wear leads to a rising failure probability.

Together these phases form the “bathtub curve,” often used to describe product lifecycle reliability.

Examples and intuition

• Human mortality: As a person ages, the probability of dying during a particular year generally increases because fewer remaining life-years remain, raising the conditional probability for that interval.
• Automobiles: Some cars fail early due to manufacturing defects (early peak). After early issues are resolved, the vehicle experiences a long period of relatively low failure risk. Later, components wear out and the failure rate rises.

Uses

The hazard rate is applied to:
* Design and safety decisions in engineering.
* Warranty and maintenance scheduling.
* Medical prognosis and survival studies.
* Risk assessment and pricing in insurance and finance.

Bottom line

The hazard rate is a compact, conditional measure of failure risk over time. Understanding its shape and behavior helps predict reliability, schedule interventions, and make data-driven decisions across engineering, healthcare, and financial contexts.