• Journal of the Korean Mathematical Society
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• v.53 no.1
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• pp.45-55
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• 2016

Let {$X_n,n{\geq}1$} be a sequence of negatively superadditive dependent random variables. In the paper, we study the strong law of large numbers for general weighted sums ${\frac{1}{g(n)}}{\sum_{i=1}^{n}}{\frac{X_i}{h(i)}}$ of negatively superadditive dependent random variables with non-identical distribution. Some sufficient conditions for the strong law of large numbers are provided. As applications, the Kolmogorov strong law of large numbers and Marcinkiewicz-Zygmund strong law of large numbers for negatively superadditive dependent random variables are obtained. Our results generalize the corresponding ones for independent random variables and negatively associated random variables.

• Communications of the Korean Mathematical Society
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• v.20 no.3
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• pp.547-562
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• 2005

The distribution of products of random variables is of interest in many areas of the sciences, engineering and medicine. This has increased the need to have available the widest possible range of statistical results on products of random variables. In this note, the distribution of the product | XY | is derived when X and Y are Student's t and Bessel function random variables distributed independently of each other.

• Bulletin of the Korean Mathematical Society
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• v.57 no.1
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• pp.51-68
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• 2020

In this paper, the author considers the nonparametric regression model with negatively orthant dependent random variables. The wavelet procedures are developed to estimate the regression function. For the wavelet estimator of unknown function g(·), the almost sure consistency is derived and the complete consistency is established under the mild conditions. Our results generalize and improve some known ones for independent random variables and dependent random variables.

• International Journal of Fuzzy Logic and Intelligent Systems
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• v.13 no.3
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• pp.215-223
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• 2013

In this paper, we present some results on weak laws of large numbers for weighted sums of fuzzy random variables taking values in the space of normal and upper-semicontinuous fuzzy sets with compact support in a separable real Banach space. First, we give weak laws of large numbers for weighted sums of strong-compactly uniformly integrable fuzzy random variables. Then, we consider the case that the weighted averages of expectations of fuzzy random variables converge. Finally, weak laws of large numbers for weighted sums of strongly tight or identically distributed fuzzy random variables are obtained as corollaries.

• Honam Mathematical Journal
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• v.32 no.1
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• pp.91-99
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• 2010

In this paper we introduce the concept of asymptotically almost negatively associated random variables in a Hilbert space and obtain the strong law of large numbers for a strictly stationary asymptotically almost negatively associated sequence of H-valued random variables with zero means and finite second moments. As an application we prove a strong law of large numbers for a linear process generated by asymptotically almost negatively random variables in a Hilbert space with this result.

• Korean Journal of Mathematics
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• v.21 no.4
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• pp.383-399
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• 2013

In this paper, we establish two types of strong law of large numbers for fuzzy random variables taking values on the space of normal and upper-semicontinuous fuzzy sets with compact support in a separable Banach space. The first result is SLLN for strong-compactly uniformly integrable fuzzy random variables, and the other is the case of that the averages of its expectations converges.

• Communications for Statistical Applications and Methods
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• v.26 no.3
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• pp.261-272
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• 2019

In this paper, we propose a procedure to build a prediction interval of the sum of dependent binary random variables over a graph to account for the dependence among binary variables. Our main interest is to find a prediction interval of the weighted sum of dependent binary random variables indexed by a graph. This problem is motivated by the prediction problem of various elections including Korean National Assembly and US presidential election. Traditional and popular approaches to construct the prediction interval of the seats won by major parties are normal approximation by the CLT and Monte Carlo method by generating many independent Bernoulli random variables assuming that those binary random variables are independent and the success probabilities are known constants. However, in practice, the survey results (also the exit polls) on the election are random and hardly independent to each other. They are more often spatially correlated random variables. To take this into account, we suggest a spatial auto-regressive (AR) model for the surveyed success probabilities, and propose a residual based bootstrap procedure to construct the prediction interval of the sum of the binary outcomes. Finally, we apply the procedure to building the prediction intervals of the number of legislative seats won by each party from the exit poll data in the $19^{th}$ and $20^{th}$ Korea National Assembly elections.

• Communications for Statistical Applications and Methods
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• v.27 no.3
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• pp.285-299
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• 2020

In many cases, we are interested in identifying independence between variables. For continuous random variables, correlation coefficients are often used to describe the relationship between variables; however, correlation does not imply independence. For finite discrete random variables, we can use the Pearson chi-square test to find independency. For the mixed type of continuous and discrete random variables, we do not have a general type of independent test. In this study, we develop a independence test of a continuous random variable and a discrete random variable without assuming a specific distribution using kernel density estimation. We provide some statistical criteria to test independence under some special settings and apply the proposed independence test to Pima Indian diabetes data. Through simulations, we calculate false positive rates and true positive rates to compare the proposed test and Kolmogorov-Smirnov test.

• Journal of the Korean Institute of Telematics and Electronics
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• v.26 no.9
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• pp.1299-1307
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• 1989

A Korean syllable is regarded as a random variable according to its probabilistic property in occurrence. A Korean syllable is divided into a 'choseong', a 'jungseong', and a 'jongseong' which are regarded as random variables. From the cumulative freaquency of a Korean syllable all possible joint probabilities and conditional probabilities are computed for the three ramdom variables. From the joint probabilities and the conditional probabilities all possible joint entropies and conditional entropies are computed for the three random varibles. Also all possible average mutual informations are calculated for the three random variables. Average mutual informatin between two random variables hss its biggest value between choseong and jungseong. Average mutual information between a random variable and other two random variables has its biggest value between jungseong and choseong-jongseong.

• Journal for History of Mathematics
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• v.32 no.3
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• pp.135-155
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• 2019

One of the reasons students have difficulty in studying probability is that they do not understand the meaning of mathematical terms precisely. One such term is a continuous random variable. Students tend not to think of the accurate definition of continuous random variables but to understand the definition of continuity of functions and the meaning of continuity in probability as equal. In this study, we try to explore the degree of pre-service teachers' understanding on the concept of continuation of functions and continuous random variables. To do this, the questionnaire items related to continuous random variables and continuity of functions were developed by experts and examined by pre-service teachers. Based on this, we make suggestions on implications for teaching and learning about continuous random variables.