# Hazard Rate

## What is 'Hazard Rate'

Hazard rate refers to the rate of death for an item of a given age (x), and is also known as the failure rate. It is part of a larger equation called the hazard function (denoted by ${\displaystyle \lambda }$), which analyzes the likelihood that something will survive to a certain point in time based on its survival to an earlier time (t). In other words, it is the likelihood that if something survives to one moment, it will also survive to the next. Hazard rate only applies to those items which cannot be repaired.

The hazard rate for any time can be determined using the following equation:

h(t) = f(t) / R(t)

f(t) is the probability density function, or the probability that the value (failure or death) will fall in a specified interval (for example, a specific year).

R(t) is the survival function, or the probability that something will survive past a certain time (t).

The hazard rate cannot be negative, and it is necessary to have a set "lifetime" on which to model the equation.

## BREAKING DOWN 'Hazard Rate'

Probability density calculates the probability of failure at any given time. For example, a person has a certainty of dying eventually: that is the probability density. As you get older, you have a greater chance of dying at a specific age, since the average failure rate is calculated as a fraction of the number of units that exist in a specific interval divided by the number of total units at the beginning of the interval.

If we were to calculate a person's chances of dying at a certain age, we would divide one year by the number of years he or she potentially has left to live. This number would grow larger each year. A person aged 60 would have a higher probability of dying at age 65 than would a person aged 30, since the person aged 30 still has many more units of time (years) left in his or her life, and the probability is less that he or she will die during one specific unit of time.

## When is Hazard Rate Used?

Hazard rate is part of a wider branch of statistics called survival analysis, which predicts the amount of time until a certain event will come to pass, specifically death or failure (as in failure of a mechanical structure). The idea is used in other branches of research under slightly different names, including reliability analysis (engineering), duration analysis (economics), and event history analysis (sociology).