Introductory Probability and Statistical Applications

Ericson, W. A.; Meyer, Paul L.

doi:10.2307/1266656

Cited by 9 publications

(17 citation statements)

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“…The standard normal factor, 15 Z , is defined as:

Z = ([], \frac{xi - E (x)}{σ})

where σ is the SD . Note that the variance of a random variable is the square of the SD of the random variable.…”

Section: Example Problemmentioning

confidence: 99%

“…What is the interlock probability of failure on demand (PFD) that we can be 95% certain that the interlock will provide? At the 95% level, the corresponding Z factor is 1.65 15 . We now compute the corresponding PFD of the interlock that corresponds to this Z factor.

Z = ([], \frac{x_{95 %} - E [Prob (T)]}{σ})

…”

Section: Example Problemmentioning

confidence: 99%

“…Tabulations of standard normal distribution are presented in most statistics books such as Meyer. 15 For this value of Z, the corresponding probability that the interlock will achieve SIL 2 performance is 4.55%. We can also compute the risk reduction that the proposed interlock design will achieve.…”

Section: Sensitivity Using Numerical Perturbation Calculationmentioning

confidence: 99%

“…At the 95% level, the corresponding Z factor is 1.65. 15 We now compute the corresponding PFD of the interlock that corresponds to this Z factor.…”

Section: Sensitivity Using Numerical Perturbation Calculationmentioning

confidence: 99%

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A novel method for fault tree uncertainty analysis using error propagation methods

Freeman

2020

Process Safety Progress

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Fault tree analysis (FTA) is a widely used methodology in the process industries. FTA is used for the development of failure mechanisms, computation of failure frequencies and the determination of the probability of failure on demand of safety systems. Much of the data used in a FTA study are uncertain. For example, the failure rate of a pump is often not known with great precision. Likewise the failure rates of instrumentation are often known only within some defined limits. The common practice, used by analysts in the quantification of a fault tree, is to use the most likely or best guess as to the needed failure rate data. The use of best guess values as data inputs to the quantification of a fault tree creates uncertainty in the computed results. This paper presents a general methodology for the determination of the impact of uncertainty on the results of a fault tree study. The general methodology is based on the mathematics of propagation of error and variance contribution analysis. For a fault tree that contains 10 event minimal cut sets, the methodology will compute the variance in the probability (or frequency) of the top event with a maximum error of 5%. For five event minimal cut sets the error in variance of the top event probability (or frequency) will be no more than 2%. An example is presented to illustrate the application of the fault tree uncertainty analysis methodology to a real world problem.

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“…The standard normal factor, 15 Z , is defined as:

Z = ([], \frac{xi - E (x)}{σ})

where σ is the SD . Note that the variance of a random variable is the square of the SD of the random variable.…”

Section: Example Problemmentioning

confidence: 99%

Z = ([], \frac{x_{95 %} - E [Prob (T)]}{σ})

…”

Section: Example Problemmentioning

confidence: 99%

Section: Sensitivity Using Numerical Perturbation Calculationmentioning

confidence: 99%

“…At the 95% level, the corresponding Z factor is 1.65. 15 We now compute the corresponding PFD of the interlock that corresponds to this Z factor.…”

Section: Sensitivity Using Numerical Perturbation Calculationmentioning

confidence: 99%

See 2 more Smart Citations

A novel method for fault tree uncertainty analysis using error propagation methods

Freeman

2020

Process Safety Progress

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“…This is the ideal case, under very favorable conditions. In statistics and probability (see [10][11][12][13][14]), the applicability of the fundamentals allows us to develop efficient models to understand and forecast the behavior of non-deterministic (random) systems or phenomena, and how they evolve over time (stochastic processes [21][22][23]): such as, gambling [8], meteorological activity, earthquakes, disintegration of atoms and radioactive nuclei, among other natural systems and phenomena. Of particular interest is the combination of the system and phenomenon of radioactive decay of nuclei represented by the so-called radioactive dice, [13,[15][16][17][18].…”

Section: Introductionmentioning

confidence: 99%

Quick counting of probability distributions for the sum of multi-sided dice: a physical and didactic approach

Sánchez-Sánchez¹,

Cortés-Pérez²,

Moreno‐Oliva³

et al. 2022

Eur. J. Phys.

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There are many physical systems (with randomness) where the performance of the probability distribution is of great importance (for example the so-called Radioactive Dice13). A relevant situation is when the value (positive integers) of the upper faces is added when rolling fair dice (unbiased), that is, balanced, regular or unshaven. It has been widely studied in the basic literature for one, two and a maximum of four dice. However, a generalization using these methods for many dice is too cumbersome to carry out. In this paper we explore a method that uses the Moment Generating Function (MGF), already studied in1, for ve six-sided fair dice with positive integer values. We carry out the generalization of this method to study the performance of the probability distribution and make a quick counting without complications for the sum of n fair dice with s sides (s 6), and real values. We also study the invariance of the distribution when we set the number of sides, that is, for s-x. Therefore, we study the distributions for the out- going total sum (random variable) of rolling generalized dice, for combinations of n multi-sided dice

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Comparison between the conventional and Bayesian approaches to evaluate measurement data

Lira

Wöger

2006

Metrologia

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Measurement data subject only to random effects can be evaluated within the frameworks of conventional as well as Bayesian statistical theory. In this paper, both viewpoints are presented and examples including Gaussian, uniform and Poisson statistics are discussed. The cases of data produced by different observers, and of quantities expressed by measurement models involving systematic effects, are also briefly touched upon. It is shown that, although in most practical cases the uncertainty intervals obtained from repeated measurements using either theory may be similar, their interpretation is completely different. Since the Bayesian approach treats random and systematic effects in the same way, the authors claim that it is more flexible and better adapted to practice than conventional theory.

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Introductory Probability and Statistical Applications

Cited by 9 publications

References 0 publications

A novel method for fault tree uncertainty analysis using error propagation methods

A novel method for fault tree uncertainty analysis using error propagation methods

Quick counting of probability distributions for the sum of multi-sided dice: a physical and didactic approach

Comparison between the conventional and Bayesian approaches to evaluate measurement data

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