Why Frequency is Not Well Defined for Engineering Systems with Nonexponential Failure Times

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Dr. Yellman proposes to define frequency as “a time-rate of events of a specified type over a particular time interval.” We review why no definition of frequency, including this one, can satisfy both of two conditions: (1) the definition should agree with the ordinary meaning of frequency, such as that less frequent events are less likely to occur than more frequent events, over any particular time interval for which the frequencies of both are defined; and (2) the definition should be applicable not only to exponentially distributed times between (or until) events, but also to some nonexponential (e.g., uniformly distributed) times. We make the simple point that no definition can satisfy (1) and (2) by showing that any definition that determines which of any two uniformly distributed times has the higher “frequency” (or that determines that they have the same “frequency,” if neither is higher) must assign a higher frequency number to the distribution with the lower probability of occurrence over some time intervals. Dr. Yellman's proposed phrase, “time-rate of events … over a particular time interval” is profoundly ambiguous in such cases, as the instantaneous failure rates vary over an infinitely wide range (e.g., from one to infinity), making it unclear which value is denoted by the phrase “time-rate of events.”

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