伯努力參數常用來代表一個實驗中成功事件的機率或是一個抽樣樣本為不良結果的機率,在現實世界中, 伯努力參數被使用在許多領域, 如系統模擬建模、統計推論、製程控制等。因此, 了解伯努力參數的本質與應用性可視為是在工程與科學領域中的一個重要問題。本研究包含四個與伯努力參數相關之議題, 其研究內容與結果如下:

1. 在第一部分中, 我們探討用於伯努力參數標準Wald信賴區間的經驗法則之適用性。在其中,本研究推導出的解析式被用於計算信賴區間涵蓋機率及解釋涵蓋機率的波動性。此外,我們得知經驗法則的必要條件為“n >μ+10σ與0<μ−10σ”被滿足時,可獲得較佳的涵蓋
機率。
2. 在第二部分中, 我們探討在製程管制內不良率p0 未知的條件下,第一階段樣本數m 與第二階段樣本數n 對於傳統不良率品質管制圖(p 管制圖)之統計特性的影響。在其中,我們推導出管制圖連串長度的條件與邊際分配之解析式, 並用於評估在p0 未知或已知情況下之管制內外的管制圖統計績效表現。此外,本研究也解釋了管制圖平均連串長度在樣本數變化下呈現波動性的原因, 以及滿足合理管制圖統計特性所需的樣本數大小。
3. 在第三部分中,我們以過去的研究為基礎,將80個包含伯努力參數相關分配在內的單變項機率分配呈現於一個使用方便的8×10矩陣上。該矩陣經由邏輯性的設計, 可讓使用者容易的找到所要的分配, 且得知該分配與其餘分配的相關性。
4. 在第四部分中,由系統觀點利用離散事件模擬方法建構包含多個伯努力參數估計值在內的急診醫療病患流程,藉以探討急診部門中病人入院策略對改善急診部門過度擁塞程度與病人提前離去現象比率之影響。由本研究,我們提供了個案醫院及急診部門關於病患入院策略的有用資訊,此外,本研究所建構之一般化模型亦可提供其他醫療單位使用。

The Bernoulli parameter, denoted as p, is the probability of success in a trial or the probability that a nonconforming result is found when a unit is sampled. In our world, the Bernoulli parameter is widely applied in various fields, for example, simulation input modeling, statistical inference, and process control. Therefore, understanding the nature and the application of the
Bernoulli parameter is an important topic in both the engineering and scientific fields. In this study, four issues related to the Bernoulli parameter are presented:

1. In the first part, we investigate the rule of thumb for the standard Wald confidence interval of a Bernoulli parameter, p. The analytical results are derived to compute the coverage probability of p and to explain the sample sizes for which the oscillation phenomenon occurs. Moreover, we correctly interpret the rule of thumb and show that satisfying a necessary condition of the rule of thumb “n >μ+ 10σ and 0<μ−10σ” can guarantee a good coverage probability.
2. In the second part, we expand on the work of Braun (1999) to consider the effects of the initial reference sample size, m, in Phase I and the on-line sampling size, n, in Phase II on the statistical performance of a conventional p chart. The conditional and marginal distributions of run length are provided, and the numerical results from in-control and out-of-control conditions with known and unknown p0 are computed for evaluation and comparison. In addition, we find that the oscillation of ARL and the right skewness of the distribution of P and P0 cause the minimal required sampling size to exceed those suggested in current textbooks.
3. In the third part, we improve upon previously published results by depicting 80 univariate probability distributions in one user-friendly ten-by-eightmatrix-format display. The figure is logically organized, and thereby allows users to easily locate a particular distribution. These 80 distributions and associated relationships provide rapid access to information that must otherwise be found through a time-consuming search of numerous sources.
4. In the last part, a discrete-event simulation approach was used tomodel the patient flow of Emergency Department’s (ED), including several estimated Bernoulli parameters, to investigate the effect of inpatient boarding on the ED efficiency in terms of the National Emergency Department Crowding Scale (NEDOCS) score and the rate of patients who leave without being seen (LWBS). The decision variable in this model was the boarderreleased-ratio, defined as the ratio of admitted patients whose boarding time is zero, to
all admitted patients. Our analysis shows that the Overcrowded+ (a NEDOCS score over 100) ratio decreased from 88.4% to 50.4%, and the rate of LWBS patients decreased from
10.8% to 8.4% when the boarder-released-ratio changed from 0% to 100%. These results show that inpatient boarding significantly impacts both the NEDOCS score and the rate
of LWBS patient, and this analysis provides a quantification of the impact of boarding on emergency department patient crowding.

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