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Fundamentals of Probability and Statistics for Reliability Analysis 21 P1: Rakesh Tung.cls Tung˙C02 July 28, 2005 14:27 Chapter 2 Fundamentals of Probability and Statistics for Reliability ∗ Analysis Assessment of the reliability of a hydrosystems infrastructural system or its components involves the use of probability and statistics. This chapter reviews and summarizes some fundamental principles and theories essential to relia- bility analysis. 2.1 Terminology In probability theory, an experiment represents the process of making obser- vations of random phenomena. The outcome of an observation from a random phenomenon cannot be predicted with absolute accuracy. The entirety of all possible outcomes of an experiment constitutes the sample space.Anevent is any subset of outcomes contained in the sample space, and hence an event could be an empty (or null) set, a subset of the sample space, or the sample space itself. Appropriate operators for events are union, intersection, and com- plement. The occurrence of events A and B is denoted as A∪ B (the union of A and B), whereas the joint occurrence of events A and B is denoted as A∩ B or simply (A, B) (the intersection of A and B). Throughout the book, the comple- ment of event A is denoted as A. When two events A and B contain no common elements, then the two events are mutually exclusive or disjoint events, which is expressed as ( A, B) =∅, where ∅ denotes the null set. Venn diagrams illustrat- ing the union and intersection of two events are shown in Fig. 2.1. When the oc- currence of event Adepends on that of event B, then they are conditional events, ∗Most of this chapter, except Secs. 2.5 and 2.7, is adopted from Tung and Yen (2005). 19 P1: Rakesh Tung.cls Tung˙C02 July 28, 2005 14:27 20 Chapter Two Figure 2.1 Venn diagrams for basic set operations. which is denoted by A| B. Some useful set operation rules are 1. Commutative rule: A∪ B = B ∪ A; A∩ B = B ∩ A. 2. Associative rule: ( A∪ B) ∪ C = A∪ (B ∪ C); ( A∩ B) ∩ C = A∩ (B ∩ C). 3. Distributive rule: A∩ (B ∪ C) = ( A∩ B) ∪ ( A∩ C); A∪ (B ∩ C) = ( A∪ B) ∩ ( A∪ C). 4. de Morgan’s rule: ( A∪ B) = A ∩ B ; ( A∩ B) = A ∪ B . Probability is a numeric measure of the likelihood of the occurrence of an event. Therefore, probability is a real-valued number that can be manipulated by ordinary algebraic operators, such as +, −, ×, and /. The probability of the occurrence of an event A can be assessed in two ways. In the case where an experiment can be repeated, the probability of having event A occurring can be estimated as the ratio of the number of replications in which event A occurs nA versus the total number of replications n, that is, nA/n. This ratio is called the relative frequency of occurrence of event A in the sequence of n replications. In principle, as the number of replications gets larger, the value of the relative P1: Rakesh Tung.cls Tung˙C02 July 28, 2005 14:27 Fundamentals of Probability and Statistics for Reliability Analysis 21 frequency becomes more stable, and the true probability of event A occurring could be obtained as nA P ( A) = lim →∞ (2.1) n n The probabilities so obtained are called objective or posterior probabilities be- cause they depend completely on observations of the occurrence of the event. In some situations, the physical performance of an experiment is prohibited or impractical. The probability of the occurrence of an event can only be estimated subjectively on the basis of experience and judgment. Such probabilities are called subjective or prior probabilities. 2.2 Fundamental Rules of Probability Computations 2.2.1 Basic axioms of probability The three basic axioms of probability computation are (1) nonnegativity: P ( A) ≥ 0, (2) totality: P (S) = 1, with S being the sample space, and (3) additivity:For two mutually exclusive events A and B, P ( A ∪ B) = P ( A) + P (B). As indicated from axioms (1) and (2), the value of probability of an event occurring must lie between 0 and 1. Axiom (3) can be generalized to consider K mutually exclusive events as K K P ( A1 ∪ A2 ∪···∪AK ) = P ∪ Ak = P ( Ak) (2.2) k=1 k=1 An impossible event is an empty set, and the corresponding probability is zero, that is, P (∅) = 0. Therefore, two mutually exclusive events A and B have zero probability of joint occurrence, that is, P ( A, B) = P (∅) = 0. Although the prob- ability of an impossible event is zero, the reverse may not necessarily be true. For example, the probability of observing a flow rate of exactly 2000 m3/s is zero, yet having a discharge of 2000 m3/s is not an impossible event. Relaxing the requirement of mutual exclusiveness in axiom (3), the probabil- ity of the union of two events can be evaluated as P ( A ∪ B) = P ( A) + P (B) − P ( A, B) (2.3) which can be further generalized as K K P ∪ Ak = P ( Ak) − P ( Ai, Aj ) k=1 k=1 i < j K + P ( Ai, Aj , Ak) −···+(−1) P ( A1, A2, ..., AK ) i < j < k (2.4) P1: Rakesh Tung.cls Tung˙C02 July 28, 2005 14:27 22 Chapter Two If all are mutually exclusive, all but the first summation term on the right-hand side of Eq. (2.3) vanish, and it reduces to Eq. (2.2). Example 2.1 There are two tributaries in a watershed. From past experience, the probability that water in tributary 1 will overflow during a major storm event is 0.5, whereas the probability that tributary 2 will overflow is 0.4. Furthermore, the probability that both tributaries will overflow is 0.3. What is the probability that at least one tributary will overflow during a major storm event? Solution Define Ei = event that tributary i overflows for i = 1, 2. From the prob- lem statements, the following probabilities are known: P (E1) = 0.5, P (E2) = 0.4, and P (E1, E2) = 0.3. The probability having at least one tributary overflowing is the probability of event E1 or E2 occurring, that is, P (E1 ∪ E2). Since the overflow of one tributary does not preclude the overflow of the other tributary, E1 and E2 are not mutually exclusive. Therefore, the probability that at least one tributary will overflow during a major storm event can be computed, according to Eq. (2.3), as P (E1 ∪ E2) = P (E1) + P (E2) − P (E1, E2) = 0.5 + 0.4 − 0.3 = 0.6 2.2.2 Statistical independence If two events are statistically independent of each other, the occurrence of one event has no influence on the occurrence of the other. Therefore, events A and B are independent if and only if P ( A, B) = P ( A)P (B). The probability of joint occurrence of K independent events can be generalized as K K P ∩ Ak = P ( A1) × P ( A2) ×···×P ( AK ) = P ( Ak) (2.5) k=1 k=1 It should be noted that the mutual exclusiveness of two events does not, in general, imply independence, and vice versa, unless one of the events is an impossible event. If the two events A and B are independent, then A, A, B, and B all are independent, but not necessarily mutually exclusive, events. Example 2.2 Referring to Example 2.1, the probabilities that tributaries 1 and 2 overflow during a major storm event are 0.5 and 0.4, respectively. For simplicity, assume that the occurrences of overflowing in the two tributaries are independent of each other. Determine the probability of at least one tributary overflowing in a major storm event. Solution Use the same definitions for events E1 and E2. The problem is to determine P (E1 ∪ E2)by P (E1 ∪ E2) = P (E1) + P (E2) − P (E1, E2) Note that in this example the probability of joint occurrences of both tributaries over- flowing, that is, P (E1, E2), is not given directly by the problem statement, as in Example 2.1. However, it can be determined from knowing that the occurrences of P1: Rakesh Tung.cls Tung˙C02 July 28, 2005 14:27 Fundamentals of Probability and Statistics for Reliability Analysis 23 overflows in the tributaries are independent events, according to Eq. (2.5), as P (E1, E2) = P (E1)P (E2) = (0.5)(0.4) = 0.2 Then the probability that at least one tributary would overflow during a major storm event is P (E1 ∪ E2) = P (E1) + P (E2) − P (E1, E2) = 0.5 + 0.4 − 0.2 = 0.7 2.2.3 Conditional probability The conditional probability is the probability that a conditional event would occur. The conditional probability P ( A| B) can be computed as P ( A, B) P ( A| B) = (2.6) P (B) in which P ( A| B) is the occurrence probability of event A given that event B has occurred. It represents a reevaluation of the occurrence probability of event A in the light of the information that event B has occurred. Intuitively, A and B are two independent events if and only if P ( A| B) = P ( A). In many cases it is convenient to compute the joint probability P ( A, B)by P ( A, B) = P (B)P ( A| B)orP ( A, B) = P ( A)P (B | A) The probability of the joint occurrence of K dependent events can be general- ized as K P ∩ Ak = P ( A1)× P ( A2 | A1)× P ( A3 | A2, A1)×···×P ( AK | AK−1, ..., A2, A1) k=1 (2.7) Example 2.3 Referring to Example 2.2, the probabilities that tributaries 1 and 2 would overflow during a major storm event are 0.5 and 0.4, respectively.
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