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Basic Reliability Mathematics
Published in Mohammad Modarres, Mark P. Kaminskiy, Vasiliy Krivtsov, Reliability Engineering and Risk Analysis, 2016
Mohammad Modarres, Mark P. Kaminskiy, Vasiliy Krivtsov
For the special case of α = β = 1, the beta distribution reduces to the standard uniform distribution. Practically, the distribution is not used as a time-to-failure distribution. On the other hand, the beta distribution is widely used as an auxiliary distribution in nonparametric classical statistical distribution estimations, as well as a prior distribution in the Bayesian statistical inference, especially when the r.v. can only range between 0 and 1: for example, in a reliability or any other probability estimate. These special applications of beta distributions are discussed in Chapter 3. Figure 2.10 shows the beta distribution pdf curves for some selected values of α and β.
Fundamentals of Applied Probability
Published in Ephraim Suhir, Human-in-the-Loop, 2018
The beta-distribution is used in statistics and in reliability engineering. If, for a certain random variable, the mean, the variance, and the minimum and maximum values are available, then it is the beta-distribution that results in the maximum entropy (uncertainty). The beta-distribution will be revisited in connection with updating reliability in the three-step concept, when the recently suggested reliability physics oriented Boltzmann–Arrhenius–Zhurkov model is sandwiched between two statistical models—Bayes formula (for technical diagnostics applications) and beta-distribution (for updating reliability, when unexpected failures have been observed).
Exposure Assessment
Published in Ted W. Simon, Environmental Risk Assessment, 2019
Another useful distribution is the beta distribution. This distribution has limits of 0 and 1. The beta distribution is often used to represent a distribution of fractional values or percentages. The distribution can also be scaled to represent an interval of values greater than 1 by setting an upper and lower limit. The shape of the beta distribution is determined by two parameters, α and β. The beta distribution is often used in Bayesian analysis to describe the proportion of success in a set of trials. In Box 4.3, the fits of the Johnson SB distribution and a scaled Beta distribution to women’s body mass are compared.
Adversarial data poisoning attacks against the PC learning algorithm
Published in International Journal of General Systems, 2020
Emad Alsuwat, Hatim Alsuwat, Marco Valtorta, Csilla Farkas
Bayesian statistics treats parameters as random variables; data is treated as fixed. For example, let θ be a parameter, and D be a dataset, then Bayes' theorem can be expressed mathematically as follows: . Since is constant (Lynch 2007), we can write Bayes' theorem in its most useful form in Bayesian update and inference as follows: It is convenient mathematically for the prior and the likelihood to be conjugate. A prior distribution is a conjugate prior for the likelihood function if the posterior distribution belongs to the same distribution as the prior (Raiffa and Schlaifer 1961). For example, the beta distribution is a conjugate prior for the binomial distribution (as a likelihood function). Equation 5 is the formula that we will use in this paper for prior to posterior update. Starting with a prior distribution , we add the count of successes,y, and the count of failures, n−y, from the dataset D (where n is total number of entries in D) to α and β, respectively. Thus, is the posterior distribution.
Repetitive inspection scheme based on the run length of test results: A Markov chain Monte Carlo approach
Published in Quality Engineering, 2020
Note that the mean of the beta distribution is a/(a + b) and the mode is (a + 1)/(a + b + 2). By changing the parameter values (a, b) in [37], we can represent a wide variety of prior distributions with different locations and scales. For illustrative purposes, we set the parameter values (a, b) = (1, 9) for each beta distribution. Thus, the prior mean is a/(a + b) = 0.10 for each of π, α, and β. We may use a different set of parameter values, such as a “non-informative” prior with (a, b) = (1, 1). For more information about non-informative prior distributions of population proportions, readers are referred to Berger (1985) and Gelman et al. (2004).
Accessibility for maintenance in the engine room: development and application of a prediction tool for operational costs estimation
Published in Ship Technology Research, 2022
Paola Gualeni, Fabio Perrera, Mattia Raimondo, Tomaso Vairo
A Beta distribution has been selected: it is a continuous probability density function defined in the range 0–1, dependent on two parameters (a and b). This distribution is widely used in Bayesian statistics because it governs the posterior probability of observing a−1 successes and b−1 failures when the prior probability is evenly distributed between 0 and 1. The formula of the Beta distribution is the following: B (a, b) normalizes the probability density function. In general, with a and b always positive: if a is assumed smaller than b, the curve will be shifted to the left; if a is equal to b, the curve will be symmetrical; and if a is greater than b, the curve will be more displaced to the right.