Editing Failure rate (section)

===Hazard rate===

A concept closely-related but different<ref name="todinov">{{cite book |last1=Todinov |first1=MT |date=2007 |title=Risk-Based Reliability Analysis and Generic Principles for Risk Reduction |chapter=Chapter 2.2 HAZARD RATE AND TIME TO FAILURE DISTRIBUTION}}</ref> to instantaneous failure rate <math>f(t)</math> is the '''hazard rate''' (or '''{{visible anchor|hazard function}}'''), <math>h(t)</math>.

In the many-system case, this is defined as the proportional failure rate of the systems ''still functioning'' at time <math>t</math> (as opposed to <math>f(t)</math>, which is the expressed as a proportion of the ''initial number'' of systems). 

For convenience we first define the reliability (or [[survival function]]) as:

:<math>R(t) = 1 - F(t)</math>

then the hazard rate is simply the instantaneous failure rate, scaled by the fraction of surviving systems at time <math>t</math>:

:<math>h(t) = \frac{f(t)}{R(t)}</math>

In the probabilistic sense, for a single system this can be interpreted as how much the [[conditional probability]] of failure time <math>T</math> within the time interval <math>t</math> to <math>t + \Delta t</math> changes, ''given that the system or component has already survived to time <math>t</math>'':

:<math>h(t) = \lim_{\Delta t \to 0^+} \frac{P(t < T \leq t + \Delta t \mid T>t)}{\Delta t}.</math>

====Conversion to cumulative failure rate====

To convert between <math>h(t)</math> and <math>F(t)</math>, we can solve the differential equation

:<math>h(t)=\frac{f(t)}{R(t)}=-\frac{R'(t)}{R(t)}</math>

with initial condition <math>R(0)=1</math>, which yields<ref name="todinov" />

:<math>F(t) = 1 - \exp{\left(-\int_0^t h(\tau) d\tau \right)}.</math>

Thus for a collection of identical systems, only one of hazard rate <math>h(t)</math>, failure probability density <math>f(t)</math>, or cumulative failure distribution <math>F(t)</math> need be defined.

Confusion can occur as the notation <math>\lambda(t)</math> for "failure rate" often refers to the function <math>h(t)</math> rather than <math>f(t).</math><ref>{{cite book| first1=Shaoping | last1=Wang |
title=Comprehensive Reliability Design of Aircraft Hydraulic System | chapter=Chapter 3.3.1.3: Failure Rate λ(t) | date= 2016}}</ref>