{"id":3913,"date":"2014-10-10T10:40:30","date_gmt":"2014-10-10T15:40:30","guid":{"rendered":"http:\/\/www.livingreliability.com\/en\/?p=3913"},"modified":"2014-11-20T08:23:37","modified_gmt":"2014-11-20T13:23:37","slug":"conditional-failure-probability-reliability-and-failure-rate","status":"publish","type":"post","link":"https:\/\/www.livingreliability.com\/en\/posts\/conditional-failure-probability-reliability-and-failure-rate\/","title":{"rendered":"Conditional failure probability, reliability, and failure rate"},"content":{"rendered":"

What is the relationship between the conditional failure probability H(t), the reliability R(t), the density function f(t), and the failure rate h(t)?<\/em><\/p>\n

In the article\u00a0\u00a0Conditional probability of failure<\/a>\u00a0we showed that the conditional failure probability H(t) is:<\/p>\n

\"H(t)=P(t\leq X\leq t+\Delta t | X>t)=\frac{P(t\leq X\leq t+\Delta t)\cap P(X>t)}{P(X>t)}\"<\/p>\n

X is the failure time. By definition the denominator is the survival or reliability function at time t<\/em>, i.e.\u00a0P(X>t) =<\/em>\u00a0R(t). The Conditional Probability of Failure is a special case of conditional probability\u00a0wherein the numerator is the intersection of two event probabilities, the first being\u00a0entirely contained within the probability space of the second, as depicted in the Venne graph:<\/p>\n

\"VenneConditionalFailureProbability\"<\/a><\/p>\n

Therefore the numerator, which is the intersection of P(t\u2264X\u2264t+\u0394t) <\/em>and \u00a0P(X>t)<\/em>\u00a0reduces to\u00a0simply\u00a0\u00a0P(t\u2264X\u2264t+\u0394t).<\/em>\u00a0Also, by expressing\u00a0P(t\u2264X\u2264t+\u0394t) <\/em>as the difference between the Cumulative Failure Probabilities calculated at t<\/em> and t+\u0394t<\/em>\u00a0<\/em>\u00a0the numerator can be expressed as the change in\u00a0Reliability over the interval \u0394t<\/em>\u00a0as:<\/p>\n

P(t\u2264X\u2264t+\u0394t) =\u00a0<\/em>F(t+\u0394t) – F(t) =\u00a0<\/em>1-R(t+\u0394t) – (1-R(t)) = R(t)-R(t+\u0394t)<\/em><\/p>\n

where the Cumulative Failure Probability F(t) and the Reliability R(t) \u00a0are complements, i.e. F(t) = 1-R(t), so that<\/p>\n

\"H(t)=\frac{R(t)-R(t+\Delta t)}{R(t)}\"<\/p>\n

We define the failure rate[1<\/a>]<\/sup> h(t) as the limit of the ratio H(t)<\/em>\/\u0394t<\/em> as \u0394t<\/em>\u21920:<\/p>\n

\"\"<\/a><\/p>\n

Differentiating \"F(t)=1-R(t)\" we have the density function f(t)<\/em>:<\/p>\n

\"f(t)=-dR(t)/dt\":<\/p>\n

Then<\/p>\n

\"h(t)=\frac{f(t)}{R(t)}\"<\/p>\n

\n\n

    \n\t
  1. [1]<\/sup><\/strong> also called hazard function or hazard rate↩<\/a><\/li><\/ol>\n","protected":false},"excerpt":{"rendered":"

    What is the relationship between the conditional failure probability H(t), the reliability R(t), the density function f(t), and the failure rate h(t)? In the article\u00a0\u00a0Conditional probability of failure\u00a0we showed that the conditional failure probability H(t) is: X is the failure time. By definition the denominator is the survival or reliability function at time t, i.e.\u00a0P(X>t) […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[89],"tags":[24,110,26],"class_list":["post-3913","post","type-post","status-publish","format-standard","hentry","category-theory-and-definitions","tag-conditional-failure-probability","tag-cumulative-failure-probability","tag-failure-rate"],"_links":{"self":[{"href":"https:\/\/www.livingreliability.com\/en\/wp-json\/wp\/v2\/posts\/3913","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.livingreliability.com\/en\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.livingreliability.com\/en\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.livingreliability.com\/en\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.livingreliability.com\/en\/wp-json\/wp\/v2\/comments?post=3913"}],"version-history":[{"count":0,"href":"https:\/\/www.livingreliability.com\/en\/wp-json\/wp\/v2\/posts\/3913\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.livingreliability.com\/en\/wp-json\/wp\/v2\/media?parent=3913"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.livingreliability.com\/en\/wp-json\/wp\/v2\/categories?post=3913"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.livingreliability.com\/en\/wp-json\/wp\/v2\/tags?post=3913"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}