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Bivariate Probability Distributions and Sampling Distributions
Published in William M. Mendenhall, Terry L. Sincich, Statistics for Engineering and the Sciences, 2016
William M. Mendenhall, Terry L. Sincich
In the preceding discussion, we defined the bivariate joint marginal and conditional probability distributions for two discrete random variables, X and Y. The concepts can be extended to any number of discrete random variables. Thus, we could define a third random variable W to each point in the sample space. The joint probability distribution p(x, y, w) would be a table, graph, or formula that gives the values of p(x, y, w), the event that the intersection (x, y, w) occurs, for all combinations of values of X, Y, and W. In general, the joint probability distribution for two or more discrete random variables is called a multivariate probability distribution. Although the remainder of this chapter is devoted to bivariate probability distributions, the concepts apply to the general multivariate case also.
Incorporating design consistency into risk-based geometric design of horizontal curves: a reliability-based optimization framework
Published in Transportmetrica A: Transport Science, 2023
Rushdi Alsaleh, Gabriel Lanzaro, Tarek Sayed
In reliability analysis, the problem formulation has two main components: the limit-state function, which is defined as a balance between demand and supply, and the random variables, which consider the design parameters’ uncertainty. The limit-state function provides negative outputs if the performance of the system is unacceptable (i.e. the system does not comply with the design requirements). The limit-state function is denoted by g(X), where X is a vector of the random variables. The output of the limit state function is defined as given by Eq.(1) and Eq.(2) (Moses 1977). The Pnc related to the limit-state function g(X) is given by Eq. (3) (Melchers and Beck 2018). where f(X) represents the random variables’ joint probability distribution function.
On the use of Bayesian networks as a meta-modelling approach to analyse uncertainties in slope stability analysis
Published in Georisk: Assessment and Management of Risk for Engineered Systems and Geohazards, 2019
The BNs are used for modelling the relationships between variables, and for capturing the uncertainty in the dependencies between these variables using conditional probabilities (van der Gaag 1996). BNs represent the joint probability distribution P(x) of a set of variables x = x1, … , xn. The joint probability distribution P(x) expresses the probability of each combination of the values of the variables in the set x. BNs enable an efficient modelling by factoring the probability distribution of each variable, given its influencing variables, into a joint probability distribution. The joint probability distribution of a network can be formulated as (Grêt-Regamey and Straub 2006)where pa(xi), is the set of values of the influencing variables of xi.
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: