「利用者:YasuakiH/Weibull distribution」の版間の差分
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Although the failure rate, <math>\lambda (t)</math>, is often thought of as the [[:en:probability|probability]] that a failure occurs in a specified interval given no failure before time <math>t</math>, it is not actually a probability because it can exceed 1. Erroneous expression of the failure rate in % could result in incorrect perception of the measure, especially if it would be measured from repairable systems and multiple systems with non-constant failure rates or different operation times. It can be defined with the aid of the [[:en:reliability function|reliability function]], also called the survival function, <math>R(t)=1-F(t)</math>, the probability of no failure before time <math>t</math>.
::<math>\lambda(t) = \frac{f(t)}{R(t)}</math>, where <math>f(t)</math> is the time to (first) failure distribution (i.e. the failure density function).▼
::<math>\lambda(t) = \frac{R(t_1)-R(t_2)}{(t_2-t_1) \cdot R(t_1)}
= \frac{R(t)-R(t+\Delta t)}{\Delta t \cdot R(t)} \!</math>
over a time interval <math>\Delta t</math> = <math>(t_2-t_1)</math> from <math>t_1</math> (or <math>t</math>) to <math>t_2</math>. Note that this is a [[:en:conditional probability|conditional probability]], where the condition is that no failure has occurred before time <math>t</math>. Hence the <math>R(t)</math> in the denominator.▼
故障率lambda(t)は、しばしば時間t以前に故障がない場合に特定の間隔で故障が発生する[[確率]]と考えられがちだが、1を超えることもあるので実際には確率ではない。故障率を誤って%で表現すると、特に修理可能なシステムや、故障率が一定でない、または動作時間が異なる複数のシステムから測定する場合において、この尺度を正しく認識できない可能性がある。故障率は、生存関数とも呼ばれる[[信頼性関数]] <math>R(t)=1-F(t)</math> （時刻 <math>t</math> 以前に故障が発生しない確率）を用いて定義できる。
::<math>\lambda(t) = \frac{f(t)}{R(t)}</math>,
ここで <math>f(t)</math> は（最初の）故障までの時間分布（すなわち故障密度関数）である。<math>t_1</math>（または <math>t</math>）から <math>t_2</math> までの時間区間 <math>\Delta t</math> = <math>(t_2-t_1)</math> において、
▲::<math>\lambda(t) = \frac{f(t)}{R(t)}</math>, where <math>f(t)</math> is the time to (first) failure distribution (i.e. the failure density function).
::<math>\lambda(t) = \frac{R(t_1)-R(t_2)}{(t_2-t_1) \cdot R(t_1)}
= \frac{R(t)-R(t+\Delta t)}{\Delta t \cdot R(t)} \!</math>
▲over a time interval <math>\Delta t</math> = <math>(t_2-t_1)</math> from <math>t_1</math> (or <math>t</math>) to <math>t_2</math>. Note that this is a [[:en:conditional probability|conditional probability]], where the condition is that no failure has occurred before time <math>t</math>. Hence the <math>R(t)</math> in the denominator.
Hazard rate and ROCOF (rate of occurrence of failures) are often incorrectly seen as the same and equal to the failure rate. {{clarify|date=April 2015}} To clarify; the more promptly items are repaired, the sooner they will break again, so the higher the ROCOF. The hazard rate is however independent of the time to repair and of the logistic delay time.
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