The total number of failures within an item population, divided by the total time expended by that population, during a particular measurement interval under stated conditions. (MacDiarmid, et al.)
Although the failure rate,
, is often thought of as the probability that a failure occurs in a specified interval given no failure before time 
, 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 reliability function, also called the survival function, 
, the probability of no failure before time .
, where 
is the time to (first) failure distribution (i.e. the failure density function) and 
.
over a time interval
from 
(or ) to 
and 
is defined as . Note that this is a conditional probability, hence the in the denominator.
The function is a CONDITIONAL probability of the failure DENSITY function. The condition is that the failure has not occurred at time .
Hazard rate and ROCOF (rate of occurrence of failures) is often incorrectly seen as the same and equal to the failure rate. And literature is even contaminated with inconsistent definitions. The hazard rate is in contrast to the ROCOF the same a failure rate. ROCOF is used for repairable systems only. In practice not many serious errors are made due to this confusion (although this statement is hard to validate...).
Although the failure rate,
, is often thought of as the probability that a failure occurs in a specified interval given no failure before time , 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 reliability function, also called the survival function, , the probability of no failure before time ., where is the time to (first) failure distribution (i.e. the failure density function) and .over a time interval
from (or ) to and is defined as . Note that this is a conditional probability, hence the in the denominator.The
function is a CONDITIONAL probability of the failure DENSITY function. The condition is that the failure has not occurred at time .Hazard rate and ROCOF (rate of occurrence of failures) is often incorrectly seen as the same and equal to the failure rate. And literature is even contaminated with inconsistent definitions. The hazard rate is in contrast to the ROCOF the same a failure rate. ROCOF is used for repairable systems only. In practice not many serious errors are made due to this confusion (although this statement is hard to validate...).
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