In reliability, we often use the “indicator” function marked I and defined by:
A.2. The factorial function
Suppose that we try to calculate the number of possible cases for combining two letters out of four. Suppose also that the order of the letters does not count (that is, A, B is identical to B, A). We will have:
– A, B
– A, C
– A, D
– B, C
– B, D
– C, D
There are six possibilities. More generally, if we combine k letters among n, we can show by induction that the possible number of cases is given by:
[A.1]
To simplify this equation, we defined the factorial function given by:
[A.2]
From equations [A.1] and [A.2], we obtain:
[A.3]
This is noted as
Let us now consider the following equation, f(n) = exp(−t). tn‒1. Let us calculate the area of this curve for n = 1. We have:
For n = 2, we have:
Integration by parts leads to: A2 = 2
For n = 3, we have:
A double integration by parts leads to: A3 = 6. More generally, it can be shown by induction that:
A.3. The complete gamma function
Thus, the Γ function defined by named the “gamma function” is a generalization of the factorial function in the space of real numbers (it is also defined for complex numbers). We can see the close link throughout this book with the Weibull distribution, for example, in the calculation of the MTTF.
A.4. Incomplete gamma function
We can see the close connection throughout this book with the Weibull distribution.
A.5. Error function
The error function, written Erf(x), was introduced by Gauss and is defined by:
References
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[AMA 06] AMARI S., “Bounds on MTBF of systems subjected to periodic maintenance”, IEEE Transactions on Reliability, vol. 55, no. 3, pp. 469–474, 2006.
[BAG 02] BAGDONAVICIUS V., NIKULIN M., Accelerated Life Models: Modeling and Statistical Analysis, Chapman & Hall, 2002.
[DEN 15] DENIS L., StaXpert report, 2015.
[GAU 07] GAUDOIN O., LEDOUX J., Modélisation aléatoire en fiabilité des logiciels, Hermès–Lavoisier, 2007.
[GIR 06] GIRAUDEAU M., “Dimensionnement des stocks de fin de vie de composants dans le cadre de leur obsolescence”, LambdaMu 15, 2006.
[MET 00] METTAS A., “Modeling & analysis for multiple stress-type accelerated life data”, Annual Reliability and Maintainability Symposium, International Symposium on Product Quality and Integrity, 2000.
[MOO 65] MOORE G., “Cramming more components onto integrated circuits”, Electronics, vol. 38, no. 8, pp. 114–117, 1965.
[NIK 07] NIKULIN M., GERVILLE-RÉACHE L., COUALLIER V., Statistiques des essais accélérés, Hermès–Lavoisier, 2007.
[RIG 00] RIGDON S.E., BASU A.P., Statistical Method for the Reliability of Repairable Systems, Wiley, 2000.
[RUS 13] RUSCHE W., HELSPER M., KRASEL S. et al., “Optimized joining technologies in IGBT modules increases the lifetime of hybrid bus inverters”, Internationaler ETG Kongress, 2013.
[SAE 96] SAE INTERNATIONAL, Guidelines and methods for conducting the safety assessment process on civil airborne systems and equipment, Standard, SAE ARP4761, 1996.
[SED 66] SEDYAKIN I.M., “About the one physical principle of theory of reliability”, Proceedings of Academy of Sciences of USSR, Technical Cybernetics, no. 3, pp. 80–87, 1966.
[USA 95] USA DEPARTMENT OF DEFENSE, Military handbook: Reliability prediction of electronic equipment, MIL-HDBK-217F, Notice 2, 1995.
[VAC 11] VACHER F., CAVET B., MIALHE F., “Power cycling fatigue and lifetime prediction of power electronic devices in space applications”, Microelectronics Reliability, vol. 59, no. 9, pp. 1985–1989, 2011.
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named the “gamma function” is a generalization of the factorial function in the space of real numbers (it is also defined for complex numbers). We can see the close link throughout this book with the Weibull distribution, for example, in the calculation of the MTTF.
