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Basics of probability and statistics
Published in Amit Kumar Gorai, Snehamoy Chatterjee, Optimization Techniques and their Applications to Mine Systems, 2023
Amit Kumar Gorai, Snehamoy Chatterjee
Although many distribution functions are developed and applied for different purposes, the Gaussian distribution, also known as the normal distribution, is the most widely used distribution function across all disciplines. The probability density function of the normal distribution is given by f(x)=12πσe−(x−μ)22σ2
Asset and Liability Management: Recent Advances
Published in George Anastassiou, Handbook of Analytic-Computational Methods in Applied Mathematics, 2019
where r1 and r2 are independent copies of r, and d__ denotes equality in distribution. The distribution is described by the following parameters: α ∈ (0,2] (index of stability), β ∈ [–1,1] (skewness parameter), μ ∈ R (location parameter), and σ ∈ [0,∞) (scale parameter). The variable is then represented as r~Sα,β (μ,σ). Gaussian distribution is actually a special case of stable distribution when α = 2, β = 0. The smaller the stability index is, the stronger the leptokurtic nature of the distribution becomes, i.e., with higher peak and fatter tails. If the skewness parameter is equal to zero, as in the case of Gaussian distribution, the distribution is symmetric. When β > 0 (β < 0), the distribution is skewed to the right (left). If β = 0 and μ = 0, then the stable random variable is called symmetric α-stable (SαS). The scale parameter generalizes the definition of standard deviation. The stable analog of variance is variation, υα, which is given by σα.
Statistical Models
Published in Anastasia Veloni, Nikolaos I. Miridakis, Erysso Boukouvala, Digital and Statistical Signal Processing, 2018
Anastasia Veloni, Nikolaos I. Miridakis, Erysso Boukouvala
The Gaussian distribution plays an important role in parametric statistics due to the relatively simple Gaussian model and its broad spectrum of applications. Indeed, in engineering and science, the Gaussian distribution is probably the most jointly used distribution for random measurements. The Gaussian distribution is also called normal distribution. The probability density function (PDF) of a Gaussian random variable X (RV) is parameterized by two parameters θ1 and θ2, which is the position parameter μ (μ ∈ ℝ) and the (squared) scale parameter σ2(σ2 > 0), respectively. The PDF of the RV x is given by: f(x;μ;σ2)=12πσe−(x−μ)22σ2,−∞<x<+∞.
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.
Ocular Artifact Suppression in Multichannel EEG Using Dynamic Segmentation and Enhanced wICA
Published in IETE Journal of Research, 2022
K. P. Paradeshi, U. D. Kolekar
Nonstationarity of the signal can be quantified at the regular time lags by measuring statistics of the signal. The signal within time lag deemed as stationary if there is no considerable variation in these statistics. Skewness, Kurtosis and discriminate are the measure statistical property of time lag provides information regarding signal. These parameters are vital in the determination of segments and artifacts present in the signal. Skewness is a measure of symmetry or, more exactly, the lack of symmetry of the distribution. A distribution, or data set, is symmetric if it looks similar to the left and right of the center point. The skewness is defined for a real signal as, For a symmetric distribution, such as Gaussian, the skewness is zero. µ = mean, σ = standard deviation. E[g(x)] represents the expected value of the function g(x) defined as, where and are discrete and continuous probability function. Generalized form of probability function is a Gaussian probability function (GPF) given as, Normalized GPF is given as, (GPF value from 0 to is 0.5)
Application of the algorithm of separating graph neural recommendation model in health information system
Published in Journal of Decision Systems, 2021
Gaussian distribution is also known as normal distribution. It is defined as an example of information will shape a dissemination, as well as through a wide margin the most notable appropriation. The conveyance gives a defined numerical capacity that can be utilised to compute the possibility for somewhat individual perception from the example space. The functional process is a plausible work that depicts how the potential gains of a variable are scattered. It is a balanced allotment wherever a huge segment of the insights bunch about the central zenith other than the probabilities expected for principles extra gone since the mean fix correspondingly in the 2 different paths. It is assumed that the scoring observation error in SGNR model obeys Gaussian distribution (Nguyen et al., 2019). So, the conditional probability density function for rij can be given: