[...]... function and the Equation (2.10) In reliability theory the survival probability is called reliability R(t)” The function R(t) corresponds to the term reliability as defined in [2.2, 2.3, 2.36, 2.38]: 2.1 Fundamentals in Statistics and Probability Theory 21 RELIABILITY is the probability that a product does not fail during a defined period of time under given functional and surrounding conditions Thus, reliability. .. used In Figure 2.4c the beam heights are determined using the relative frequency, as can be seen on the percent scale for the ordinate The division of the time axis into classes and the assignment of failure times to the individual classes is called classification In this process information is lost, since a certain amount of failures is assigned to one frequency independent of the exact failure time in. .. basic reliability observations and for the scope of this book, the definition out of Equation (2.26) is sufficient This equation will be used in the following because of its clarity Axiomatic Definition of Probability (Kolmogoroff 1933) In axiomatic definition “probability” is not defined in a strict sense In modern theory, “probability” is seen much more as a basic principle that fulfils certain axioms... human beings 2.1.1.4 Failure Rate To describe the failure behaviour with the failure rate λ(t), the failures at the point in time t or in a class i are not divided by the sum of total failures, as for the relative frequency in Section 2.1.1.1, but rather are divided by the sum of units still intact: λ( t ) = Failures (at the point in time t or in class i ) sum of units still intact (at the point in time... these random events The term probability is of particular importance and will be described in various ways in the following Classical Definition of Probability (Laplace 1812) The first contemplations concerning probability were made by gamblers interested in possible odds and where it is optimal to gamble at high stakes To answer the question “how probable” it is that a certain event A occurs in a game... definition in Equation (2.24) is not universally valid This equation is only applicable when it is not possible for an infinite amount of events to occur and when every possible result is equally likely In general, this is adequate for gambling In technical reality, however, the failure possibilities normally occur in varying amounts Statistical Definition of Probability (von Mises 1931) For a random... shown in Figure 2.14, which results by connecting the beam midpoints with straight lines The sum of failures and the sum of the intact units in each class i or at any point in time t always add up to 100% The survival probability R(t) is thus the complement to the failure probability F(t) R (t ) = 1 − F (t ) (2.12) With Equation (2.12) the histogram in Figure 2.14 can also be determined by reflecting... values described in Section 2.1.2, further values are used in the realm of reliability engineering to characterize reliability data • MTTF (mean time to failure), • MTTFF (mean time to first failure) and 2.1 Fundamentals in Statistics and Probability Theory 31 MTBF (mean time between failure), • failure rate λ und failure quota q, • percent (%), per mill (‰), parts per million (ppm) and • Bq lifetime... already survived up to this point The failure rate is determined by dividing the number of failures per time period by the sum of units still intact The failure quota q can serve as an estimation of the failure rate λ In contrast to the failure rate, the failure quota specifies the relative change in an observed time interval q= failures in a time interval initial quantity ⋅ interval size (2.23) If, for... specimen size of 50 units within one hour, then the failure quota is 1 q = 0,1 (“10% per hour”) [2.8] h Percent, Per Mill and PPM In the realm of reliability engineering many circumstances are represented proportionally, such as the failure density, the failure probability or the reliability The representation of these values is most commonly given in: 2.1 Fundamentals in Statistics and Probability Theory . will be presented and explained. The Weibull distribution, which is mainly and commonly used in mechanical engineering, will be explained in detail. 1 Introduction 5 System Reliability Assurance Constructive: Optimal. B. Bertsche, Reliability in Automotive and Mechanical Engineering. VDI-Buch, doi: 10.1007/978-3-540-34282-3_2, © Springer-Verlag Berlin Heidelberg 2008 8 2 Fundamentals of Statistics and Probability. remains our task to avoid faults if possible“ Sir Karl R. Popper B. Bertsche, Reliability in Automotive and Mechanical Engineering. VDI-Buch, doi: 10.1007/978-3-540-34282-3_1, © Springer-Verlag