[...]... 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. .. 2.27 Bx lifetime 2.1.4 Definition of Probability As described in the previous sections, the failure times of components and systems can be seen as random variables The terms and laws of mathematical probability theory can be applied to 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... the reliability The representation of these values is most commonly given in: 2.1 Fundamentals in Statistics and Probability Theory • percent: • per mill: • ppm: 33 quantity out of 1 hundred, i.e 1 out of 100 = 1 %, quantity out of 1 thousand, i.e 1 out of 1,000 = 1 ‰ and quantity out of 1 million, i.e 1 out of 1.000.000 = 1 ppm Bx lifetime The Bx lifetime describes the point in time at which x % of. .. estimation and not with a definition Trying to develop an all inclusive probability theory on the basis of Equation (2.26) resulted in degrees of acceptance and mathematical difficulties which could not be solved However, for 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... quota specifies the relative change in an observed time interval q= failures in a time interval initial quantity ⋅ interval size (2.23) If, for example, 5 units fail out of a test 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... of human deaths 80 years 100 16 2 Fundamentals of Statistics and Probability Theory 2.1.1.2 Distribution Function or Failure Probability In many cases, the number of failures at a specific point in time or in a specific interval is not of interest but rather, how many components in total have failed up to a time or until a certain interval is reached This question can be answered with a histogram of. .. number of failures = nA n (2.2) can be 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... 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 of gambling, Laplace and Pascal determined the following definition: 34 2 Fundamentals of Statistics and Probability Theory Probability P( A ) = number of cases favorable to A number of all possible... 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 t or in class i ) (2.13) Figure 2.18 shows the histogram of the failure rates and the function of the empirical failure rate λ *(t) for the trial run in Figure 2.4 It can be seen, that the failure rate in the last... independent of the exact failure time in the interval Through the classification, each failure within a certain class is assigned the value of that class’s mean However, a loss of information is compensated by a win in overview The amount of classes is not always simple to determine If the classes are chosen to be too large, then too much information is lost In an extreme case, there is only one beam, which of . 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. distribution, one of the most adopted in mechanical engineering will be explained. Section 2.3 combines component reliability with system reliability with the help of Boolean theory. The Boolean theory. concerning their product reliability. No one wants to be confronted with a lack of reliability in their product. Often, these kinds of statements are kept under strict secrecy. An interesting