Editing Mean time between failures (section)

== Mathematical description ==
The MTBF is the expected value of the random variable <math>T</math> indicating the time until failure. Thus, it can be written as<ref name="birolini">Alessandro Birolini: ''Reliability Engineering: Theory and Practice''. Springer, Berlin 2013, {{ISBN|978-3-642-39534-5}}.</ref>
: <math>\text{MTBF} = \mathbb{E}\{T\} = \int_0^\infty tf_T(t)\, dt</math>
where <math>f_T(t)</math> is the [[probability density function]] of <math>T</math>. Equivalently, the MTBF can be expressed in terms of the [[reliability function]] <math>R_T(t)</math> as
: <math>\text{MTBF} = \int_0^\infty R(t)\, dt </math>.
The MTBF and <math>T</math> have units of time (e.g., hours).

Any practically-relevant calculation of the MTBF assumes that the system is working within its "useful life period", which is characterized by a relatively constant [[failure rate]] (the middle part of the "[[bathtub curve]]") when only random failures are occurring.<ref name="lienig" /> In other words, it is assumed that the system has survived initial setup stresses and has not yet approached its expected end of life, both of which often increase the failure rate.

Assuming a constant failure rate <math>\lambda</math> implies that <math>T</math> has an [[exponential distribution]] with parameter <math>\lambda</math>. Since the MTBF is the expected value of <math>T</math>, it is given by the reciprocal of the failure rate of the system,<ref name="lienig" /><ref name="birolini" />
: <math>\text{MTBF} = \frac{1}{\lambda}</math>.

Once the MTBF of a system is known, and assuming a constant failure rate, the [[probability]] that any one particular system will be operational for a given duration can be inferred<ref name="lienig" /> from the [[reliability function]] of the [[exponential distribution]], <math>R_T(t) = e^{-\lambda t}</math>. In particular, the probability that a particular system will survive to its MTBF is <math>1/e</math>, or about 37% (i.e., it will fail earlier with probability 63%).<ref>{{cite web|title= Reliability and MTBF Overview|url= http://www.vicorpower.com/documents/quality/Rel_MTBF.pdf |publisher= Vicor Reliability Engineering |access-date=1 June 2017}}</ref>