• Journal of the Korean Statistical Society
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• 제23권2호
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• pp.483-498
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• 1994

Suppose that a stationary process ${X_t}$ has a marginal distribution whose support consists of sufficiently large integers. We are concerned with some analogous law of large numbers for such distribution function F. In particular, we determine a weak law of large numbers for maximum queueing length in $M/M\infty$ system. We also present a limiting behavior for the maxima based on AR(1) process with binomial thining and poisson marginals (INAR(1)) introduced by E. Mckenzie. It turns out that the result of AR(1) process is the same as that of $M/M/\infty$ queueing process in limit when we observe the queues at regularly spaced intervals of time.

• 대한수학회논문집
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• 제33권2호
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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.

• 한국컴퓨터정보학회논문지
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• 제21권7호
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• pp.85-92
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• 2016

The law of large numbers, central limit theorem, and connection among binomial distribution, normal distribution, and statistical estimation require dynamics of continuous visualization for students' better understanding of the concepts. During this visualization process, the differences and similarities between statistical probability and mathematical probability that students should observe need to be provided with the intermediate steps in the converging process. We propose a visualization method that can integrate intermediate processes and results through Excel. In this process, students' experiences with dynamic visualization help them to perceive that the results are continuously changed and extracted from multiple situations. Considering modeling as a key process, we developed a classroom exercise using Excel to estimate the population mean and standard deviation by using a sample mean computed from a collection of data out of the population through sampling.

• 한국통계학회:학술대회논문집
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• 한국통계학회 2003년도 추계 학술발표회 논문집
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• pp.253-256
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• 2003

In ecological studies, animal science, or entomology, the variance of count is considered to have the power of the mean relationship with the mean count as Taylor (1961) presented his famous 'Taylor's Power Law'. In this talk, we are going to review the development of TPL and its extension toward pest management sampling scheme. Different estimation methods are compared. Quasilikelihood approach is suggested to incorporate covariate information. Possible extensions will be discussed.

• 대한수학회논문집
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• 제25권1호
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• pp.119-128
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• 2010

Let ${X_n,\;n\geq1}$ be a sequence of independent identically distributed (i.i.d.) random variables (r.vs.), defined on a probability space ($\Omega$,A,P), and let ${N_n,\;n\geq1}$ be a sequence of positive integer-valued r.vs., defined on the same probability space ($\Omega$,A,P). Furthermore, we assume that the r.vs. $N_n$, $n\geq1$ are independent of all r.vs. $X_n$, $n\geq1$. In present paper we are interested in asymptotic behaviors of the random sum $S_{N_n}=X_1+X_2+\cdots+X_{N_n}$, $S_0=0$, where the r.vs. $N_n$, $n\geq1$ obey some defined probability laws. Since the appearance of the Robbins's results in 1948 ([8]), the random sums $S_{N_n}$ have been investigated in the theory probability and stochastic processes for quite some time (see [1], [4], [2], [3], [5]). Recently, the random sum approach is used in some applied problems of stochastic processes, stochastic modeling, random walk, queue theory, theory of network or theory of estimation (see [10], [12]). The main aim of this paper is to establish some results related to the asymptotic behaviors of the random sum $S_{N_n}$, in cases when the $N_n$, $n\geq1$ are assumed to follow concrete probability laws as Poisson, Bernoulli, binomial or geometry.

• 식물병연구
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• 제21권4호
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• pp.268-272
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• 2015

마늘 세균점무늬병은 국내에서 제주지역 난지형 마늘에서 Pseudomonas syringae pv. morri에 의해 발생하는 세균병해로 주요 병해 중 하나이다. 본 연구는 자연로그를 취한 이항분산에 대한 관측분산의 회귀모델인 binary power law를 이용하여 포장 내 병 발생의 공간분포 특성을 분석하였다. 회귀식의 기울기(b) 값이 1.361로 기준치인 1보다 커 세균점무늬병은 공간적으로 집중분포하고 있었다. 축차표본조사법은 평균이병주율($p_m$)의 추정과 요방제수준($p_t$) 이상 또는 이하 여부를 판단할 수 있도록 개발하였다. 본 연구결과는 기존의 조사지점수 고정 조사 방법에 비해 조사비용 면에서 더 효율적인 동시에 조사결과의 정확도를 더 높일 수 있을 것으로 판단된다.

• 한국수학교육학회지시리즈A:수학교육
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• 제43권4호
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• pp.391-398
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• 2004

It is well-known in the literature that the general term of a sequence can be represented by a linear combination of binomial coefficients. The theorem and its known proofs are not easy for highschool students to understand. In this paper we prove the theorem by a pictorial method and by a very short and easy inductive method to make the problem easy and accessible enough for highschool students.

• 한국응용곤충학회지
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• 제51권2호
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• pp.91-97
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• 2012

온주밀감에서 귤녹응애, Aculops pelekassi의 분산지수와 분포양상, 표본조사시 적정 표본수에 대하여 조사하였다. 귤녹응애는 집중분포를 하고 있었으며, 분산지수는 Taylor's power law가 Iwao's patchiness regression보다 더 잘 설명하고 있었다. Taylor's power law의 상수를 이용하여 고정 정확도 수준에서 열매 표면 $cm^2$당 누적충수에 따라 조사를 중지할 수 있는 표본조사법을 만들었다. 경제적인 표본조사를 위하여 Kono-sugino의 경험적 이항모델을 개발하였으며, 이항모델을 이용하면 귤녹응애가 $cm^2$당 12마리 이상 발생한 열매 비율을 이용하여 평균밀도를 추정할 수 있었다 : $ln(m)=4.61+1.23ln[-ln(1-p_{12})]$. 최적의 tally threshold를 결정하기 위하여 추정평균에 대한 분산을 계산한 결과 tally threshold가 12일 때 추정평균의 분산이 적었으며, 발생과율 0.1~0.5의 범위에서 분산의 변동이 거의 없어 다른 tally threshold에 비해 높은 정확도로 평균을 추정할 수 있었다. 적정 표본수를 결정하기 위하여 계층표본조사법을 이용하여 분석한 결과 고정 정확도 0.25수준에서 감귤원당 적정 조사 나무수는 13주였으며, 나무당 조사 열매수는 5개, 열매당 2지점에서 $cm^2$당 귤녹응애수 조사가 바람직하였다(총 130표본).

• 한국안전학회지
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• 제33권3호
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• pp.77-82
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• 2018

This paper aims to analyze the accident based on changing patterns of traffic culture index. For this purpose, this paper particularly focuses on classifying the traffic culture patterns and developing the traffic accidents using panel count data model. The main results are as follows. First, the traffic culture patterns are divided into 'increasing', 'decreasing' and 'other' patterns. The null hypotheses that the number of accident are the same over patterns are rejected. Second, 4 fixed effect negative binomial models which are all statistically significant are developed. Third, the regions with 'increasing' pattern are analyzed to be mostly the counties, and to demand the traffic law enforcement. Fourth, the regions with 'decreasing' pattern are evaluated to be mainly the districts and to require such the traffic culture as turn signal usage. Finally, the regions with 'other' pattern are analyzed to be mostly the cities and to ask for enhancing the level of traffic culture.

• 농업과학연구
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• 제47권3호
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• pp.615-624
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• 2020

An effective sampling method is necessary to monitor potato tuber moths (Phthorimaea operculella) because they are the biggest concern in potato-cultivating areas. In this study, a sequential sampling method was developed based on the results of field surveys of potato tuber moths in South Korea. Potato tuber moths were collected in fields cultivating potatoes at six sites, and their spatial distribution was investigated using the Taylor power law. The optimal sampling size and cumulative number of potato tuber moths in traps to stop sampling were determined based on the spatial distribution pattern and mean density of the collected potato tuber moths. Finally, the developed sampling method was applied to propose a control action, and its sampling efficiency was compared with that of the traditional sampling method using a binomial distribution. The potato tuber moths tended to aggregate; the optimal number was approximately 5 - 16 traps for sampling, and the number varied with the mean density of potato tuber moths according to the sampling sites. In addition, one, two, and three sites might require the following actions: Continued sampling, control, and no control, respectively. Sampling with the binomial distribution showed the minimum sample size was 12 when considering the economic threshold level. Here, we propose an effective sampling method that can be applied for future monitoring and field surveys of potato tuber moths in South Korea.