Editing Failure rate (section)

===Failures over time===

[[File:Exponential distribution cdf.svg|thumb|300px|Cumulative distribution function for the [[exponential distribution]], often used as the cumulative failure function <math>F(t).</math>]]

To accurately model failures over time, a '''cumulative failure distribution''', <math>F(t)</math> must be defined, which can be any [[cumulative distribution function]] (CDF) that gradually increases from <math>0</math> to <math>1</math>. In the case of many identical systems, this may be thought of as the fraction of systems failing over time <math>t</math>, after all starting operation at time <math>t=0</math>; or in the case of a single system, as the [[probability]] of the system having its failure time <math>T</math> before time <math>t</math>:

:<math>F(t) = \operatorname{P}(T\le t).</math>

As CDFs are defined by integrating a [[probability density function]], the '''failure probability density''' <math>f(t)</math> is defined such that: 

[[File:Exponential pdf.svg|thumb|right|300px|Exponential probability functions, often used as the failure probability density <math>f(t)</math>.]]

:<math>F(t)=\int_{0}^{t} f(\tau)\, d\tau \!</math>

where <math>\tau</math> is a dummy integration variable. Here <math>f(t)</math> can be thought of as the ''instantaneous failure rate'', i.e. the fraction of failures per unit time, as the size of the time interval <math>\Delta t</math> tends towards <math>0</math>:

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