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Advanced Project Planning and Risk Managemen
Published in Timothy J. Havranek, Modern Project Management Techniques for the Environmental Remediation Industry, 2017
The second important point is that the sum of the probabilities of rolling the numbers 2 through 12 is one. This is a fundamental requirement for probabilities of a set of events which are mutually exclusive and collectively exhaustive. Mutually exclusive means that one and only one possible outcome can occur in a given trial. This is obviously true for a single roll of the dice where the sum of the two die can result in only one of the numbers 2 through 12. Collectively exhaustive means that there are no possible outcomes other than those in our set. This is also true for a pair of dice (i.e., it is not possible to roll the number 13). The second fundamental concept of probability theory for mutually exclusive and collectively exhaustive events is mathematically stated in Equation 10.2: () ∑i=1nP(ei)=1
Probabilistic considerations
Published in Mark Aschheim, Enrique Hernández-Montes, Dimitrios Vamvatsikos, Design of Reinforced Concrete Buildings for Seismic Performance, 2019
Mark Aschheim, Enrique Hernández-Montes, Dimitrios Vamvatsikos
Conditional probability and mutual exclusivity provide us with our second major tool in modeling and solving probability problems: the theorem of total probability. This theorem tells us that in order to measure, e.g., the weight of a building, which is rather impossible to weigh on a scale, we need only to measure the weights of its individual components and sum them up. We simply have to be careful not to miss any component and not to double-count. In mathematical terms, let complex event A be composed by the mutually exclusive and collectively exhaustive events E1, E2, …, Ek. Then:
Master-slave Robots, Optimal Control, Quantum Mechanics and Informationlds
Published in Harish Parthasarathy, Electromagnetics, Control and Robotics, 2023
Quantum mechanics is based on the following postulates : ( 1) The state of the system at any time is described by a density operator ρ in a Hilbert space ℌ. By a density operator, we mean that it is positive semidefinite and its trace is unity. The state ρ is said to be pure if it has rank one. In that case, we can write ρ = |ψ〉 〈ψ| where |ψ〉 is a unit vector in ℌ. The state ρ plays the role of a probability measures in classical probability theory while the Hibert space ℌ plays the role of the sample space. ( 2) An event is an orthogonal projection operator P in ℌ. The probability of the event P occurring when the system is in the state ρ is defined as Tr(ρP). The Boolean s algebra of events in classical probability is replaced by P(ℌ), the lattice of orthogonal projections in ℌ. Just as in classical probability theory, we talk of a set of mutually exclusive and collectively exhaustive events (i.e., pairwise disjoint and whose union is the entire sample space), the analog in quantum probability is a spectral resolution of the identity, i.e., a countable set {Pi} of orthogonal projections in ℌ such that PiPj = Piδij and ∑i Pi = I. The analogue of an observable in the quantum theory is a Hermitian operator X in the Hilbert space. Suppose we take a classical mechanical system described by generalized coordinates (q1,,...,,qn) and generalized momenta (p1...pn), then we have the Poisson bracket relation {qi,pj}=δij
Data-driven stochastic model for train delay analysis and prediction
Published in International Journal of Rail Transportation, 2023
The stochastic matrix with elements pij is central to any Markov chain model in Eq. (3). This matrix is called the one-step transition matrix because each element represents the probability of transition from the current state i (row i) to the next state j (column j) in one time step. Each row of the one-step transition matrix constitutes the sample space of the next states for the current state, and therefore each row has a set of mutually exclusive and collectively exhaustive events. Hence, the sum of the elements at each row equals 1; i.e., . One of the widely used methods to compute the transition probabilities is the maximum likelihood estimate in which the number of observed transitions nij from state i to state j and the total number of observations in state i before the transition ni are used, where . The one-step transition probability from current state i to next state j is computed as follows:
Probabilistic life-cycle seismic resilience assessment of aging bridge networks considering infrastructure upgrading
Published in Structure and Infrastructure Engineering, 2020
This formulation is based on the total probability theorem (Ang & Tang, 2007). The bridge damage combinations represent a set of Ns mutually exclusive and collectively exhaustive events, where Ns is the number of all the possible permutations of the entries in vector s. In particular, r represents the outcomes of the resilience measure random variable and vector ηh collects the parameters affecting the seismic hazard expressing the information on seismic intensities and seismogenic sources, such as epicenter location and earthquake magnitude. The resilience measure distribution also depends on the earthquake occurrence time t0, since the detrimental effects of aging and the benefits of infrastructure upgrading have a critical impact on the damage combination probabilities.
Lifetime seismic resilience of aging bridges and road networks
Published in Structure and Infrastructure Engineering, 2020
Luca Capacci, Fabio Biondini, Andrea Titi
Based on the ordered selection with repetitions of the damage states sb for each bridge in a given road system, network damage combinations can be associated with the following integer index s where Nb is the number of bridges in the network and Ns,b is the number of possible damage states for the b-th bridge. Consistently with the set of possible damage states of a single bridge, all the possible network damage combinations represent a set of mutually exclusive and collectively exhaustive events. In particular, the event s = 1 refers to the undamaged network condition (sb = 0, ∀b) and the combination associated with all bridges suffering the most severe damage is s = Ns, which is the total number of possible network damage combinations