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Probability
Published in John Bird, Bird's Basic Engineering Mathematics, 2021
An independent event is one in which the probability of an event happening does not affect the probability of another event happening. If 5 transistors are taken at random from a batch of transistors and the probability of a defective transistor, p1, is determined and the process is repeated after the original 5 have been replaced in the batch to give p2, then p1 is equal to p2. Since the 5 transistors are replaced between draws, the second selection is said to be with replacement.
Getting started with supply chain analytics
Published in Peter W. Robertson, Supply Chain Analytics, 2020
At a very basic level, probability assigns the likelihood of an outcome (an ‘event’) between the numerical values of 0 and 1, where 0 signifies the event will not happen and 1 signifies the event will always happen. Additionally, all possible results of an event must add up to 1. Table 3.4 provides an example where the frequencies of differing customer demand levels over a period of one year (365 days) are shown. As can be seen, the individual instance probabilities are all between 0 and 1, thus: 0≤Pevent≤1
Reliability Statistics
Published in Ali J Jamnia, Khaled Atua, Executing Design for Reliability within the Product Life Cycle, 2019
Poisson distribution describes the probability of occurrence of certain numbers of an event in a continuous domain such as fixed time distance or length. It can be used in a manufacturing facility to model the expected number of defects in a given process. The main assumption is that the occurrences of these events are totally independent of each other, i.e., the occurrence of this event is random and has a constant rate. Also, events occur at certain intervals, and no two events happen at the same instant. The probability of a certain number of events in a fixed time interval is given by the following equation: Pkeventsinanintervalt=e−λλkk!
Sequential fusion estimation for Markov jump systems with heavy-tailed noises
Published in International Journal of Systems Science, 2023
Hui Li, Liping Yan, Yuqin Zhou, Yuanqing Xia, Xiaodi Shi
Notation: denotes the n-dimensional Euclidean space. The set of positive integers is denoted by . denotes a probability density function (pdf) for a random variable. denotes the probability of a random event. denotes the Gaussian pdf with mean μ and covariance Σ, denotes the Student's t pdf with mean , scale matrix , and degrees of freedom (dof) ν.
Ocean waves time-series generation: minimum required artificial wave time-series for wave energy converter analysis
Published in Journal of Marine Engineering & Technology, 2023
Mohammad Reza Tabeshpour, Navid Belvasi
The normal distribution, also known as the Gaussian distribution, is a commonly used continuous PDF in probability theory. This is because many natural phenomena can be modelled using this distribution. The main reason for this is the role of the normal distribution in the central limit theorem. According to the central limit theorem, the probability distribution of diverse phenomena with finite mean and variance tends to fit a normal distribution, and as the number of samples increases, the probability density function (PDF) fits the Gaussian distribution even better (Cox 2006). For instance, in experiments where a certain value is measured, multiple variables such as visual error, measurement instrument error, and environmental conditions error can affect the measurement errors, but with multiple measurements, these errors still follow a normal distribution and are scattered around a constant value (Lyon 2014). The formulation of the Gaussian distribution is given by equation (28). Where or the mean determines the distribution location and as the standard deviation (rotational variance) determines the distribution scale.
Vulnerability of transmission towers under intense wind loads
Published in Structure and Infrastructure Engineering, 2022
Edgar Tapia-Hernández, David De-León-Escobedo
Seventy angle-section coupon tests were reported in the study with theoretical stress equal to Fy Theor= 345 MPa (50 ksi; 3,515 kg/cm2). According to the results, the actual yielding stress exceeds the theoretical stress (Fy Theor< Fy Actual). With the purpose of establishing a reference for the possible variation of the material strength, a statistical study was carried out. The process of fitting distribution involves the use of certain statistical methods in order to estimate the distribution parameters based on the sample data. Here, a lognormal, normal, and kernel tri-weight distributions were considered between other potential probability distributions (Figure 9a) to estimate a proper statistical distribution for the actual characteristics of the material. A normal distribution is a type of continuous probability distribution for a real-valued random variable. Kernel and log-normal distribution had been previously commented. A χ2 goodness-of-fit test was performed to select the better fit (see Table 2); according to the results, a log-normal distribution is consistent enough with the available overstrength material factors and it was selected in this study. Following the procedure defined by Equations 1 and 2, parameters were equal to λ = 0.1285 and ξ = 0.0553.