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Basic Stochastic Mathematics
Published in Ning Zhang, Chongqing Kang, Ershun Du, Yi Wang, Analytics and Optimization for Renewable Energy Integration, 2019
Ning Zhang, Chongqing Kang, Ershun Du, Yi Wang
A probability distribution is a mathematical function that provides the probabilities of occurrence of different possible outcomes in an experiment. The sum of all the probabilities for all possible outcome is equal to 1: () ∑i=-∞+∞pi=1,or∫x=-∞+∞p(x)=1.
Spectral Density
Published in David C. Swanson, ®, 2011
The central limit theorem states that regardless of the probability distribution of an independent random variable Xn, the sum of a large number of identical but statistically independent random variables (each of the same probability distribution) gives a random variable Y which tends to have a Gaussian probability distribution. This may seem like a remarkable assumption, but it is not from a detailed mathematical perspective. () Y=1N[X1+X2+X3+…+XN]
Probability and statistics
Published in Alan Jeffrey, Mathematics, 2004
Random variables can be discrete in nature, like the number of telephone calls to an office during a given period, which must be an integral number. Conversely, random variables can also be continuous, as occurs when the actual resistance of a nominally 1000 ohm resistor produced by a production line is measured. Thus both discrete and continuous random variables must be considered. This in turn leads to the introduction of discrete and continuous probability distributions, and to the study of their consequences when related to randomly occurring events. The most commonly occurring discrete probability distributions are the binomial and Poisson distributions, while the most important continuous distribution is certainly the normal or Gaussian distribution, all of which are considered in this chapter.
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
At this point, three probability distributions were computed before selecting a log-normal distribution: beta, kernel, and log-normal as depicted in Figure 7. In general, probability distributions describe the dispersion of the value of a random variable. The beta distribution is applied to model the behavior of random variables limited to intervals of finite length in a wide variety of disciplines. Kernel density estimation is a non-parametric way to estimate the probability density function of a random variable. Whereas in a log-normal distribution the random variable takes only positive real values. A χ2 goodness of fit test was performed (see Table 1); according to the results, the log-normal distribution represents a most reasonable estimation of the considered values and, therefore, as discussed above, it is the select distribution for the study. In Table 1, OF stands for observed frequency and EF refers to the expected frequency. Note that OF Actual, EF beta, EF LN and EF Kernel have units of percentage of total occurrences, and χ2 beta, χ2 LN, and χ2 Kernel are non-dimensional.
A system optimisation design approach to vehicle structure under frontal impact based on SVR of optimised hybrid kernel function
Published in International Journal of Crashworthiness, 2021
Xianguang Gu, Wei Wang, Liang Xia, Ping Jiang
Through the analysis of the load-carrying path, the load-carrying capacity of path C is of great significance to the energy absorption and collision acceleration of vehicles. It is the main load-bearing component of vehicle body structure when the front collision occurs. Therefore, the thickness of eight components is chosen as design variables in this study, as shown in Figure 8 (taking into account the symmetry of the structure). Table 2 displays the probabilistic distribution and variation of the variables, and these variables are assumed to be continuous. Table 2 shows the probability distribution and variation range of design parameters of variables, assuming that the fluctuations of these variables are continuous. The normal distribution is a very important probability distribution in statistics and it is also the widely used continuous distribution in engineering practice [28–30]. As long as the sample data is sufficient, the normal distribution can be adopted to describe its probability characteristics. Thus the design variables are treated as normal distributions in RBDO and robust optimisation in this study, whose coefficient of variation (COV()) is set as 5% from typical manufacturing and assembly tolerance [29]. The variations of design variables are chosen according to possible design range.