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Useful Maths
Index
This Maths section is a necessary part of Reliability Engineering, giving you the tools required to perform Reliability Calculations. The maths detailed here have been kept as simple as possible.

The list outlined below describes some of the most commonly used equations within Reliability Engineering:

        Reliability Function
        Unreliability Function (or Cumulative Distribution Function [CDF])
        Probability Density Function (PDF)
        Failure Rate Function
        Hazard Rate Function
        Mean Time to Failure (MTTF)
        Conditional Reliability
        Useful Relationships




Reliability Function
The Reliability Function, R(t), represents the probability of failure prior to some point in time, represented by (t). The Reliability Function is usually used when reliabilities are being calculated.



Unreliability Function (or Cumulative Distribution Function [CDF])
F(t) is the probability that a failure will occur before time t and is called the Cumulative Distribution Function [CDF].
F(t) = 1-R(t). F(t) is also referred to as the unreliability function and can be thought of representing the probability of failure prior to some time, t.

The Unreliability Function is usually used when probabilities of failure are being calculated.



Probability Density Function (PDF)
The PDF, given the symbol f(t), describes the shape of the failure distribution and is defined by:



Given the PDF, f(t):



Which simply means that the Reliability Function R(t) and the Unreliability Function F(t) represent the area under the curve defined by f(t).




Failure Rate Function
The rate at which a failure occurs in the time interval t1 to t2, is called the Failure Rate Function and is given the symbol l(t):





Hazard Rate Function
The Hazard Rate, h(t) (or Instantaneous Hazard Rate) is the Failure Rate as the time interval t1 to t2 approaches zero. More simply it provides a measure of the change in survivor rate per unit change in time:



Beware! Some books and references use the symbol l(t) when referring to either the Failure Rate Function or the Hazard Rate Function and in truth the terms are often used synonymously. However, you might think of the Failure Rate as an average rate of failure over time, where as the Hazard Rate is the rate of failure at any particular time.




Mean Time to Failure (MTTF)
The MTTF is the expected time to failure and by definition:



However, it can also be expressed as:





Conditional Reliability
The reliability of a system for a period of t hours given that it has already operated for a period of T hours is referred to as the Conditional Reliability. It might apply when considering the reliability of an item following a warranty period.

 Reliability for a period t given that the system has operated for a period T:



For the Exponential or Constant Failure Rate Model:






Useful Relationships
The Reliability Function if you know the Hazard Rate or Failure Rate:

   or   


The Reliability Function if you know the Hazard Rate or Failure Rate and the Probability Density Function:

   or   
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