• 대한수학회지
• /
• 제59권6호
• /
• pp.1185-1201
• /
• 2022

Feller introduced an unfair-fair-game in his famous book [3]. In this game, at each trial, player will win 2^k yuan with probability p_k = 1/2^kk(k + 1), k ∈ ℕ, and zero yuan with probability p₀ = 1 - Σ^∞_k=1 p_k. Because the expected gain is 1, player must pay one yuan as the entrance fee for each trial. Although this game seemed "fair", Feller [2] proved that when the total trial number n is large enough, player will loss n yuan with its probability approximate 1. So it's an "unfair" game. In this paper, we study in depth its convergence in probability, almost sure convergence and convergence in distribution. Furthermore, we try to take 2^k = m to reduce the values of random variables and their corresponding probabilities at the same time, thus a new probability model is introduced, which is called as the related model of Feller's unfair-fair-game. We find out that this new model follows a long-tailed distribution. We obtain its weak law of large numbers, strong law of large numbers and central limit theorem. These results show that their probability limit behaviours of these two models are quite different.

• Geomechanics and Engineering
• /
• 제39권1호
• /
• pp.1-11
• /
• 2024

Geotechnical parameter estimation is critical to the design, performance, safety, and cost and schedule management in Tunnel Boring Machine projects. Since these parameters vary within a certain range, relying on mean values for evaluation introduces significant risks to the project. Due to the non-homogeneous characteristics of geological formation, data may not exhibit a normal distribution and the presence of outliers might be deceptive. Therefore, the use of reliable analyses and simulation models is inevitable in the course of the data evaluation process. Advanced modeling techniques enable comprehensive analysis of the project data and allowing to model the uncertainty in geotechnical parameters. This study involves using Monte Carlo Simulation method to predict probabilistic distributions of field data, and therefore, establish a basis for designs and in turn to minimize project risks. In the study, 166 sets of geotechnical data Obtained from 35 boreholes including Standard Penetration Test, Limit Pressure, Liquid Limit, and Plastic Limit values, which are mostly utilized parameters in estimating project requirements, were used to estimate the geotechnical data distribution of the study field. In this context, firstly, the data was subjected to multi-parameter linear regression and variance analysis. Then, the obtained equations were implemented into a Monte Carlo Simulation, and probabilistic distributions of the geotechnical data of the field were simulated and corresponding to the 90% probability range, along with the minimum and maximum values at the 5% probability levels presented. Accordingly, while the average SPT N30 value is 42.86, but the highest occurrence rate is 50.81. For Net Limit Pressure, the average field data is 17.07 kg/cm², with the maximum occurrence between 9.6 kg/cm² and 13.7 kg/cm². Similarly, the average Plastic Limit value is 22.32, while the most probable value is 20.6. The average Liquid Limit value is 56.73, with the highest probability at 54.48, as indicated in the statistical data distribution. Understanding the percentage distribution of data likely to be encountered in the project allows for accurate forecasting of both high and low probability scenarios, offering a significant advantage, particularly in ordering TBM requirements.

• ETRI Journal
• /
• 제28권3호
• /
• pp.301-310
• /
• 2006

In mobile communications, a class of variable-complexity algorithms for convolutional decoding known as sequential decoding algorithms is of interest since they have a computational time that could vary with changing channel conditions. The Fano algorithm is one well-known version of a sequential decoding algorithm. Since the decoding time of a Fano decoder follows the Pareto distribution, which is a heavy-tailed distribution parameterized by the channel signal-to-noise ratio (SNR), buffers are required to absorb the variable decoding delays of Fano decoders. Furthermore, since the decoding time drawn by a certain Pareto distribution can become unbounded, a maximum limit is often employed by a practical decoder to limit the worst-case decoding time. In this paper, we investigate the relations between buffer occupancy, decoding time, and channel conditions in a system where the Fano decoder is not allowed to run with unbounded decoding time. A timeout limit is thus imposed so that the decoding will be terminated if the decoding time reaches the limit. We use discrete-time semi-Markov models to describe such a Fano decoding system with timeout limits. Our queuing analysis provides expressions characterizing the average buffer occupancy as a function of channel conditions and timeout limits. Both numerical and simulation results are provided to validate the analytical results.

• 대한수학회논문집
• /
• 제14권3호
• /
• pp.627-633
• /
• 1999

Let the given distribution $\pi$ have a log-concave density which is proportional to exp(-V(x)) on $R^d$. We consider a Markov chain induced by the method Gibbs sampling having $\pi$ as its in-variant distribution and prove geometric ergodicity and the functional central limit theorem for the process.

• 소성∙가공
• /
• 제27권2호
• /
• pp.115-122
• /
• 2018

Most domestic and international standards on the forming limit diagram (FLD) including ISO 12004-2, use a 'position-dependent method,' which determines the forming limit from a strain distribution measured on the specimen after necking or fracture. However, the position-dependent method has inherent problems such as the incidence of asymmetry of a strain distribution, the estimation of missing data near fracture, the termination time of test, and the deformation due to the new stress equilibrium after a fracture, which is blamed for causing sometimes a significant lab-to-lab variation. The 'time-dependent method,' which is anticipated to be a new international standard for evaluating the forming limit, is expected to greatly improve these intrinsic disadvantages of the position-dependent method. It is because the time-dependent method makes it possible to identify and accurately determine the forming limit, just before the necking point from the strain data as continuously measured in a short time interval. In this study, we propose a new time-dependent method based on a Gaussian fitting of strain acceleration with the introduction of 'normalized correlation coefficient.' It has been shown in this study that this method can determine the forming limit very stably and gives a higher value, which is in comparison with the results of the previously studied position-dependent and time-dependent methods.

• The Korean Journal of Ecology
• /
• 제6권1호
• /
• pp.45-54
• /
• 1983

A distribution map of black pine, Pinus thunbergii, in south Korea was prepared through field surveys. According to Mirov(1967), the range of black pine is limited in east Asia, Japan and Korea; its northern limit is at about 41。34' north and its southern limit on Takara island, south Kyushu, at 29。 north. According to the present map, its northern limit coincided with the isopath of warmth index 100 by Yim(1977). The density of pine in grid mesh, about 4 km * 4 km, was higher at costal area than that in inland area, and it was also higher at sea side slope than the opposite slope of the same mountain. This suggests that the ecological distribution of black pine is greatly affected by salt content of the soil or the optimal range of summed temperature for the pine growth.

• Communications for Statistical Applications and Methods
• /
• 제3권1호
• /
• pp.243-255
• /
• 1996

The number of neutron signals from a neutral particle beam(NPB) at the detector, without any errors, obeys Poisson distribution, Under two assumptions that NPB scattering distribution and aiming errors have a circular Gaussian distribution respectively, an exact probability distribution of signals becomes a Poisson-power function distribution. In this paper, we show that the error rate in simple hypothesis testing for the limiting Poisson-power function distribution is not zero. That is, the limit of ${\alpha}+{\beta}$ is zero when Poisson parameter$\kappa\rightarro\infty$, but this limit is not zero (i.e., $\rho\ell$>0)for the Poisson-power function distribution. We also give optimal decision algorithms for a specified error rate.

• 소성∙가공
• /
• 제20권1호
• /
• pp.36-41
• /
• 2011

Meso-scale or micro-scale forming of sheet metal parts has been recently considered as one of the important forming technologies with growing demand on meso/micro products for electric or medical devices. Experimental investigation on the cylindrical meso-cup drawing with hemispherical punch is carried out to examine the limit drawing ratio and thickness distribution of drawn cups. The working parameters chosen in this study are blank diameter, die-corner radius and blankholding force. It is found from the experiments that the limit drawing ratio of 2.4 can be achieved in the case of hemispherical cup drawing and uniform thickness distribution in wider region can be obtained compared with the results of conventional cup drawing.

• Journal of the Korean Data and Information Science Society
• /
• 제17권1호
• /
• pp.221-231
• /
• 2006

A chi-squared test of multivariate normality is suggested which is oriented for detecting deviations from elliptical symmetry. We derive the limiting distribution of the test statistic via a central limit theorem on empirical processes. A simulation study is conducted to study the accuracy of the limiting distribution in finite samples. Finally, we compare the power of our method with those of other popular tests of multivariate normality under a non-normal distribution.

• Geomechanics and Engineering
• /
• 제24권4호
• /
• pp.399-406
• /
• 2021

It is a key issue in the tunnel design to evaluate the stability of the excavation face. Two efficient analytical models in the context of the limit equilibrium method (LEM) and the limit analysis method (LAM) are used to carry out the deterministic calculations of the safety factor. The safety factor obtained by these two models agrees well with that provided by the numerical modelling by FLAC 3D, but consuming less time. A simple probabilistic approach based on the Mote-Carlo Simulation technique which can quickly calculate the probability distribution of the safety factor was used to perform the probabilistic analysis on the tunnel face stability. Both the cumulative probabilistic distribution and the probability density function in terms of the safety factor were obtained. The obtained results show the effectiveness of this probabilistic approach in the tunnel design.