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Probability Distributions
Published in Alan R. Jones, Probability, Statistics and Other Frightening Stuff, 2018
We will find (assuming that we look) that the Gamma Distribution is typically used in Queueing Theory models, some meteorological forecasts, and certain sectors of the financial services industry. It models the time required for a number of random events to occur where we know the average number of events to expect in a period of time. Internet resources cite a number of varied uses including: Levels of rainfallThe size of loan defaults or aggregate insurance claimsThe flow of items through manufacturing or distribution facilities and processesDemand on web ServersDisk drive failuresDemand on telecommunication centres The Chi-Squared Distribution is used typically to measure the ‘goodness of fit’ of sample data to a distribution on the assumption that the modelling error is Normally Distributed (see Chapter 6). The Sample Variance of a Normal Distribution can be shown to follow as Chi-Squared Distribution (Vose, 2008, p.613).
Queuing Theory
Published in Paul J. Fortier, George R. Desrochers, Modeling and Analysis of Local Area Networks, 1990
Paul J. Fortier, George R. Desrochers
where NSi denotes the number of elements in bin i due to the random sample, and NDi is the number in bin i due to the hypothesized distribution. The basis of this test is that the statistic of Equation (6-117) has a chi-square distribution. The degree of freedom of the chi-square distribution is defined as one less than the number of sample bins minus the number of parameters in the hypothesized distribution. () m=k−1-numberofparameters
Nanooncology: Molecular Imaging, Omics, and Nanoscale Flow-Mediated Medicine Tumors Strategies
Published in Sarhan M. Musa, Nanoscale Flow, 2018
Tannaz Farrahi, Tri Quang, Keerthi Srivastav Valluru, Suman Shrestha, George Livanos, Yinan Li, Aditi Deshpande, Michalis Zervakis, George C. Giakos
The term degree of freedom is defined as the number of terms in the final calculation of a statistical problem, herein the chi-square distribution, that we can vary freely. The distribution of the local variance image of the remaining area has no significant contribution as this region appears homogenous; thus, its variance equals to values near zero. The mean of the χ2 distribution is n, its variance is 2n, and its mode equals to n − 2 [98]. As the degrees of freedom increase, the chi-square function approximates the Gaussian one.
Particle Emission Fluctuation in Turbulent Diffusion Flames
Published in Combustion Science and Technology, 2022
Kesong Zhang, Yonghui Liu, Ruixia Chu, Wanyou Huang
The fitting results, summarized in Table 1, show that Pearson’s correlation coefficients between measured and fitted data are more than 0.9 for the primary reference fuels, indicating a high correlation between the measured and fitted curves. Additionally, the degree of freedom for the fitting curve is higher for a higher content of iso-octane in the fuels. The degree of freedom for a chi-square distribution represents the number of standard normal random variables. A higher degree of freedom implies more random events. The formation and oxidation of particles in turbulent diffusion flame are considered random events. High iso-octane binary fuel, which is a high sooting trend fuel, shows a high degree of freedom in certain turbulence.
Reasoning support for predicting requirement change volatility using complex network metrics
Published in Journal of Engineering Design, 2022
Phyo Htet Hein, Elisabeth Kames, Cheng Chen, Beshoy Morkos
In observing the distribution of the evaluation measures, it is found that they do not follow normality for all RCV classes. Therefore, performance comparison cannot be achieved through a typical analysis of variance – which assumes normality of the data – rather, the Friedman test, a nonparametric test without distributional assumptions, is performed to identify which model differs from which others (Salkind 2006). The test statistic is approximated by the chi-squared distribution with degrees of freedom and corresponding p-value (Salkind 2006). If the p-value is less than the significance level, the alternative hypothesis indicates that at least one differing model exists (Salkind 2006). When the Friedman test’s null hypothesis is rejected, statistical tests namely post hoc multiple comparison tests are used to identify which model is different. Here, we use the Nemenyi test, to determine which models different significantly if their mean ranks differ by at least one Nemenyi’s test statistic, critical difference (CD). The Friedman post hoc test for multiple comparisons and its test statistic is also performed. Additionally, the p-values of the test are adjusted using Shaffer’s static procedure. If the corresponding p-value between any two models is less than the significant level, the alternative hypothesis indicates that they are different in performance.
Seating preferences in highly automated vehicles and occupant safety awareness: A national survey of Chinese perceptions
Published in Traffic Injury Prevention, 2020
Bingbing Nie, Shun Gan, Wentao Chen, Qing Zhou
The respondent perspectives regarding which situations might lead to failing to wear seat belts were more prevalent to be designated as the rear seats (Q31, Mean = 4.37, S.D. = 0.91) rather than the front seats (Q32, Mean = 2.90, S.D. = 1.17). Furthermore, respondents from less developed cites stated that it was more common to observe unrestrained occupants in cars (Supplementary material Tables A3 and A4). Respondents expressed the situations that they were likely to be unrestrained were being in the rear seat (48.5%), in a short trip (41.4%), cruising at low speed (29.4%), forgetting (25.7%); while only 22.1% chose to be restrained all the time (Q32). The behavior of wearing seat belts all the time was predicted by binary logistic regression (Table 1). The p-value of a Chi-squared distribution is calculated by the HL test (χ2 (6, N = 1,018) = 15.8, p = 0.05). The model estimated that the means of transport (p < 0.01) and the city tier (p < 0.01) were the critical determinants of seat belt wearing behaviors. The commuters who were used to taking public transport (e.g., bus) had a lower possibility (Odds Ratio = 0.79) to wear seat belts. Meanwhile, respondents from lower tier (less developed) cities also designated a similar tendency (Odds Ratio = 0.80).