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Correlation and functions of random variables
Published in Andrew Metcalfe, David Green, Tony Greenfield, Mahayaudin Mansor, Andrew Smith, Jonathan Tuke, Statistics in Engineering, 2019
Andrew Metcalfe, David Green, Tony Greenfield, Mahayaudin Mansor, Andrew Smith, Jonathan Tuke
The row sums give us the distribution of these companies by world class in the right hand margin. When the distribution of one variable is obtained from a bivariate distribution in this way it is referred to as a marginal distribution. The column totals give us the marginal distribution of companies by size. To demonstrate the concept of a bivariate probability mass distribution we will divide the numbers of companies in categories by their total (298) to obtain relative frequencies, and define a probability mass function with probabilities equal to these relative frequencies. A bivariate pmf has the form PXY(x,y)=P(X=x∩Y=y).
Review of Probability Theory
Published in Harry G. Perros, An Introduction to IoT Analytics, 2021
When studying a problem, it is possible that we may be able to calculate the joint probability distribution of a number of random variables, but we may be only interested in the probability distribution of one of these random variables. A probability distribution calculated from a joint distribution is called the marginal distribution.
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
We can see that both the marginal distribution and conditional distribution can be deduced from the joint distribution. However, constructing the joint distribution of two or more random variables is not a trivial task because of their complex dependencies. We provide more deep insight into this issue in the next chapter.
Stochastic simulation of daily runoff in the middle reaches of the Yangtze river based on SVM-Copula model
Published in Systems Science & Control Engineering, 2019
Pingyi Wang, Jie Zhang, Meili Wang, Yuchun Liang, Jian Li
The parameters of SVM-Copula daily runoff series stochastic model can be divided into two categories: marginal distribution function parameters and the Copula function parameters. The parameters of the marginal distribution function can generally be obtained by the moment method, the linear moment method or the maximum likelihood method. The correlation index method is usually used to determine the Copula function parameters. The parameter θ can be calculated by the relationship with the Kendall rank correlation coefficient. The relationships between θ and are as follows: GH Copula function: Clayton Copula function: AMH Copula function: Frank Copula function:
Comparative analysis of PM2.5 pollution risk in China using three-dimensional Archimedean copula method
Published in Geomatics, Natural Hazards and Risk, 2019
The risk factors H, E, and V of the PM2.5 pollution are all continuous random variables, and the marginal distribution of the univariate was obtained based on the parameter estimation. The considered marginal distribution for the single variable includes Beta, Exponential, Extreme value, Gamma, Logistic, Log-logistic, Lognormal, Normal, Generalized extreme value, and Weibull distribution. The optimal marginal distribution was established by the A-D test (Table 3). In Table 3, Weibull and Gamma are the main types of the univariates, especially for H and V, with Lognormal and Generalized extreme value distribution followed. The E indicator takes a high proportion of Lognormal distribution. The only E in YeR and ScR is Exponential distribution.
Future Challenges of Particulate Matters (PMs) Monitoring by Computing Associations Among Extracted Multimodal Features Applying Bayesian Network Approach
Published in Applied Artificial Intelligence, 2022
Amani Abdulrahman Albraikan, Jaber S. Alzahrani, Noha Negm, Lal Hussain, Mesfer Al Duhayyim, Manar Ahmed Hamza, Abdelwahed Motwakel, Ishfaq Yaseen
The p (X, Y) shows joint probability distribution of X and Y. However, p(X) and p (Y) indicate the marginal distribution of X and Y, respectively. The relevant Gaussian distribution of co-variance matrix variables X1, X2, X3, … . Xn (Xiao et al. 2016) can be computed as: