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博碩士論文 etd-0801114-140626 詳細資訊
Title page for etd-0801114-140626
論文名稱
Title
機率與統計理論在精算考試之應用
Application of Probability and Statistics Theory in Actuarial Exam
系所名稱
Department
畢業學年期
Year, semester
語文別
Language
學位類別
Degree
頁數
Number of pages
248
研究生
Author
指導教授
Advisor
召集委員
Convenor
口試委員
Advisory Committee
口試日期
Date of Exam
2014-07-24
繳交日期
Date of Submission
2014-09-01
關鍵字
Keywords
分佈函數、精算考試、統計模擬、存活分析、貝氏定理、事件機率
survival analysis, distribution function, event probability, simulation method, actuarial exam, Bayes’ theorem
統計
Statistics
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中文摘要
本文主要目的在於探討 SOA 及 CAS 精算考試試題中與機率統計相關的理論與應
用包含四個章節: 機率概論、單維隨機變數、多維隨機變數及損失模型,並介紹解題方
法中常使用到的定義及定理。內容主要包含: 機率概論、事件機率、條件機率、獨立事
件、古典機率、貝氏定理、單維與多維隨機變數的機率分佈、分佈函數、期望值、變異
數、動差與動差生成母函數、變數變換、中央極限定理、順序統計量。最後將介紹與損
失模型相關的參數化統計方法、存活分析、統計模擬。透過本文範例的整理與分析,希
望能夠增進讀者對於機率與統計學習的效果。並且對於精算領域中與機率與統計相關理
論有更深一層的了解。
Abstract
This thesis investigates the methods of solving probability and statistics prob- lems
in SOA & CAS actuarial exams. These exam problems are classified as 4 main topics.
The topics include general probability, univariate probability distributions, multivariate
probability distributions and loss models. We will introduce definitions and theorems used
to solve problems. In Chapter 2 we introduce general probabil- ity including event
probability, conditional probability, independent events, classical probability and Bayes’
theorem. In Chapter 3&4 we discuss one-dimensional and multi-dimensional random
variable including probability distribution, distribution function, expected value, variance,
moments, moment generating function, variable transformation, CLT and order statistics.
In Chapter 5 we introduce loss models including survival analysis and simulation
method. The aim of this thesis to en- hance the readers to better understanding of
probability and statistics theory and applications in actuarial exams.
目次 Table of Contents
目錄
論文審定書 ................................................................... i
致謝....................................................................... ii
摘要 ....................................................................... iii
Abstract ................................................................. iv
表次 viii
圖次 x
第一章 前言 1
第二章 機率概論 3
2.1 事件的機率 ........................................................... 3
2.1.1 樣本空間與事件 ................................................ 3
2.1.2 古典機率 .............................................. 11
2.1.3 機率公理與機率性質 ........................................ 14
2.2 條件機率及獨立事件 .................................................. 15
2.2.1 條件機率 .................................................. 15
2.2.2 獨立事件 ..................................................... 18
2.3 貝氏定理 ............................................................ 26
第三章 隨機變數 33
3.1 機率分佈 ............................................................ 34
3.1.1 隨機變數 .................................................. 34
3.1.2 機率分佈 ..................................................... 35
3.1.3 常見的離散型及連續型隨機變數 .................................42
3.2 分佈函數及其函數 ....................................................51
3.3 期望值變異數及標準差 ................................................58
3.4 動差及動差生成母函數 ................................................76
3.5 常見的離散型及連續型隨機變數其 m.g.f.、期望值及變異數 ...............80
3.6 變數變換 ............................................................81
第四章 多維隨機變數 87
4.1 機率分佈 ............................................................88
4.1.1 機率分佈 .....................................................88
4.1.2 條件機率分佈 ................................................ 102
4.2 分佈函數及隨機變數的獨立性 ......................................... 114
4.3 期望值、變異數及共變異數 ........................................ 117
4.3.1 期望值、變異數及共變異數 ................................. 117
4.3.2 條件期望值與條件變異數 ...................................... 131
4.3.3 總機率法則及其應用 .......................................... 143
4.4 動差及動差母函數 ................................................... 151
4.5 變數變換 ........................................................... 154
4.6 中央極限定理 ....................................................... 157
4.7 順序統計量 ...................................................... 165
第五章 損失模型 170
5.1 存活分析 ........................................................ 170
5.1.1 Kaplan-Meier、Nelson-Aalen 估計法 ........................ 174
5.1.2 估計存活函數與累積危險函數的變異數及信賴區間 ............. 185
5.1.3 Kaplan-Meier 近似法 ...................................... 194
5.2 參數化統計方法 ..................................................... 198
5.2.1 動差法 ................................................... 198
5.2.2 最大概似法 .................................................. 202
5.2.3 卡方檢定 .................................................... 213
5.3 統計模擬 ........................................................... 218
5.3.1 逆變換法 ................................................. 218
5.3.2 拔靴法 ...................................................... 223
參考文獻 231
A 名詞介紹 232
索引 234
參考文獻 References
Bowers, N.L., Gerber, H.U., Hickman, J.C., Jones, D.A., and Nesbitt, C.J. (1997). Actu-
arial Mathematics, 2nd edition. Society of Actuaries, Schaumburg, Illinois.
Casella, G. and Berger, R.L. (2002). Statistical Inference, 2nd edition. CA: Duxbury,
Pacific Grove.
Dickson, D., Hardy, M. and Waters, H. (2009). Actuarial Mathematics for Life Contingent
Risks, 1st edition. Cambridge University Press, New York.
Hogg, R.V. and Craig, A.T. (1983). Introduction to Mathematical Statistics, 4th edition.
Macmillan Publishing Company Inc, New York.
Hogg, R.V. and Tanis, E.A. (2009). Probability and Statistical Inference, 8th edition.
Pearson Education, Inc., New York.
Klugman, S.A., Panjer, H.H. and Willmot, G.E. (2010). Loss Models: From Data to
Decisions, 4th edition. John Wiley & Sons, Inc., New York.
Ostaszewski, K. (2010). Study Manual for Exam P, 14th edition. ASM Publications.
Ross, S. (2009). A First Course in Probabaility, 8th edition. Pearson Education, Inc., New
York.
Weishaus, A. (2012). Study Manual for Exam C, 14th edition. ASM Publications.
黃文璋(2003). 數理統計,第一版,華泰,臺北。
黃文璋(2010). 機率論。第二版,華泰,臺北。
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