• East Asian mathematical journal
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• v.34 no.1
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• pp.59-67
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• 2018

We present new recurrence formulas for the raw and central moments of a compound binomial random variable. Our approach involves relating two compound binomial random variables that have parameters with a difference of 1 for the number of trials, but which have the same parameters for the success probability for each trial. As a consequence of our recursions, the raw and central moments of a binomial random variable are obtained in a recursive manner without the use of Stirling numbers.

• The Pure and Applied Mathematics
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• v.29 no.1
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• pp.31-35
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• 2022

In the note, by virtue of Abel's theorem and Abel's limit theorem in the theory of power series, the author provides three proofs for a sum of an alternating series involving central binomial numbers.

• Journal of the Korean Data and Information Science Society
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• v.14 no.2
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• pp.385-391
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• 2003

A sequence of n Bernoulli trials which violates the constant success probability assumption is termed as "Poisson trials". In this paper, the recurrence formula for the r-th central moment of number of successes with n Poisson trials is derived. Romanovsky's method, based on the differentiation of characteristic function, is used in the derivation of recurrence formula for the central moments of conventional binomial distribution. Romanovsky's method is applied to that of Poisson trials in this paper. Some central moment calculation results are given to compare the central moments of Poisson trials with those of conventional binomial distribution.

• Bulletin of the Korean Mathematical Society
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• v.54 no.1
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• pp.225-242
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• 2017

A family of $Ap{\acute{e}}ry$-like series involving reciprocals of central binomial coefficients is studied and it is shown that they represent transcendental numbers. The structure of such series is further examined in terms of finite combinations of logarithms and arctangents with arguments and coefficients belonging to a suitable algebraic extension of rationals. Monotonicity of certain quotients of weighted binomial sums which arise in the study of competitive cheap talk models is established with the help of a continuous extension of the discrete model at hand. The monotonic behavior of such quotients turns out to have important applications in game theory.

• International Journal of Highway Engineering
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• v.18 no.4
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• pp.83-92
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• 2016

PURPOSES : The purpose of this study was to develop safety performance functions (SPFs) that use zero-inflated negative binomial regression models for urban intersections in central business districts (CBDs), and to compare the statistical significance of developed models against that of regular negative binomial regression models. METHODS : To develop and analyze the SPFs of intersections in CBDs, data acquisition was conducted for dependent and independent variables in areas of study. We analyzed the SPFs using zero-inflated negative binomial regression model as well as regular negative binomial regression model. We then compared the results by analyzing the statistical significance of the models. RESULTS : SPFs were estimated for all accidents and injury accidents at intersections in CBDs in terms of variables such as AADT, Number of Lanes at Major Roads, Median Barriers, Right Turn with an Exclusive Turn Lane, Turning Guideline, and Front Signal. We also estimated the log-likelihood at convergence and the likelihood ratio of SPFs for comparing the zero-inflated model with the regular model. In he SPFs, estimated log-likelihood at convergence and the likelihood ratio of the zero-inflated model were at -836.736, 0.193 and -836.415, 0.195. Also estimated the log-likelihood at convergence and likelihood ratio of the regular model were at -843.547, 0.187 and -842.631, 0.189, respectively. These figures demonstrate that zero-inflated negative binomial regression models can better explain traffic accidents at intersections in CBDs. CONCLUSIONS : SPFs that use a zero-inflated negative binomial regression model demonstrate better statistical significance compared with those that use a regular negative binomial regression model.

• Journal of the Korean Society of Safety
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• v.30 no.2
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• pp.70-76
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• 2015

This study deals with the traffic accident of circular intersections. The purpose of the study is to develop the accident models by traffic violation type. In pursuing the above, this study gives particular attention to analyzing various factors that influence traffic accident and developing such the optimal models as Poisson and Negative binomial regression models. The main results are the followings. First, 4 negative binomial models which were statistically significant were developed. This was because the over-dispersion coefficients had a value greater than 1.96. Second, the common variables in these models were not adopted. The specific variables by model were analyzed to be traffic volume, conflicting ratio, number of circulatory lane, width of circulatory lane, number of traffic island by access road, number of reduction facility, feature of central island and crosswalk.

• Communications of the Korean Mathematical Society
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• v.33 no.2
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• pp.605-618
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• 2018

The main goal of this paper is to study an extension of random summations of independent and identically distributed random variables when the number of summands in random summation is a partial sum of n independent, identically distributed, non-negative integer-valued random variables. Some characterizations of random summations are considered. The central limit theorems and weak law of large numbers for extended random summations are established. Some weak limit theorems related to geometric random sums, binomial random sums and negative-binomial random sums are also investigated as asymptotic behaviors of extended random summations.

• Journal of the Korean Society of Safety
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• v.32 no.5
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• pp.129-134
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• 2017

This study aims to develop the traffic accident models by circular intersection type using count data model. The number of accident, the number of fatal and injured persons(FSI), and EPDO are calculated from the traffic accident data of TAAS. The circular intersection accident models are developed through Poisson and negative binomial regression analysis. The main results of this study are as follows. First, the null hypotheses that there are differences in the number of traffic accidents, FSI and EPDO by type of circular intersections are rejected. Second, the scale of intersection(median, large), number of approach road, mean width and length of exit road, area of the circulating roadway and central island are selected as factors influencing the number of traffic accidents, FSI and EPDO in rotary. Third, the scale of intersection(median), guide signs(limited speed, direction, roundabout), number of approach road, entry angle, area of the intersection and central island are adopted as factors influencing the number of traffic accidents, FSI and EPDO in roundabout. Finally, transferring from rotary to roundabout could be expected to make the accident decrease.

• International Journal of Highway Engineering
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• v.14 no.2
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• pp.145-155
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• 2012

At present, Seoul City is putting the bus exclusive lane system into practice according to mass transit revitalization policy. Starting with the installation of roadside bus exclusive lane in the past, at present, even the road sections for central- lane bus exclusive lane system are on the increase. The purpose of this research is to analyze the factors giving impacts on bus accident on central bus exclusive lane and roadside bus exclusive lane. In case of the central bus exclusive lane, the 6 variables, such as the number of bus routes, number of access & entrance to central lanes patterns, whether the stop line of central lanes retreats or not, separated distance between the stop line of central lanes and crosswalks, traffic volume, and number of bus routes stopping at bus stops on reversible lanes, were found to have a significant influence on bus accidents. In case of roadside bus exclusive lane sections, the four variables such as the number of right-turn bus routes, whether to be chronic illegal parking & stopping, time for the walk signal, and forms of land use, etc. were found to have a significant influence on bus accident.

• Journal of Korean Society of Transportation
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• v.35 no.2
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• pp.160-168
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• 2017

In the past 10 years, the accidents caused by drowsy driving have occupied about 23% of all traffic accidents in Korea expressway network and this rate is the highest one among all accident causes. Unlike other types of accidents caused by speeding and distraction to the road, the accidents by drowsy driving should be managed differently because the drowsiness might not be controlled by human's will. To reduce the number of accidents caused by drowsy driving, researchers previously focused on the spot based analysis. However, what we actually need is a segment (link) and occurring time based analysis, rather than spot based analysis. Hence, this research performs initial effort by adapting link concept in terms of drowsy driving on highway. First of all, we analyze the accidents caused by drowsy in historical accident data along with their road environments. Then, links associate with driving time are analyzed using digital tachograph (DTG) data. To carry this out, negative binomial regression models, which are broadly used in the field, including highway safety manual, are used to define the relationship between the number of traffic accidents on expressway and drivers' behavior derived from DTG. From the results, empirical Bayes (EB) and potential for safety improvement (PSI) analysis are performed for potential risk segments of accident caused by drowsy driving on the future. As the result of traffic accidents caused by drowsy driving, the number of the traffic accidents increases with increase in annual average daily traffic (AADT), the proportion of trucks, the amount of DTG data, the average proportion of speeding over 20km/h, the average proportion of deceleration, and the average proportion of sudden lane-changing.