戴爾指數是由熵所推導而得的。在這篇論文中，我們研究熵及其相關特性。我們用了條件熵的特性去研究戴爾指數可藉由組組之間結合之特性，並且去探討瓦瑟斯坦度量之間的關聯性。最後我們將戴爾指數與基尼指數用在探討新冠肺炎病例分佈不均的例子上。

Theil index is derived from entropy. In this thesis, we study entropy
and some of its properties. We use properties of conditional entropy to
study Theil index composed by groups, and explore the connection with
Wasserstein metric. We apply Theil index and Gini index to an example
showing unevenness of COVID-19 cases.

Contents
1 Introduction 1
2 Entropy 4
2.1 Partitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.2 Entropy of a Partition . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.3 Conditional Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.4 Entropy of a Measure-Preserving Transformation . . . . . . . . . . 22
2.5 Kolmogorov-Sinai theorem . . . . . . . . . . . . . . . . . . . . . . . 28
2.6 Bernoulli shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3 Theil and Gini index 34
3.1 Theil index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.2 Gini index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.3 Wasserstein metric . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
4 Application of Theil and Gini index 44
Reference 53

References
[1] Tudorel Andrei, Bogdan Oancea, Peter Richmond, Gurjeet Dhesi, and
Claudiu Herteliu, Decomposition of the Inequality of Income Distribution by
Income Types—Application for Romania, Entropy 2016, 19, 430 : pp. 9
[2] P.Walters, Gradudate Text in Mathematics: An Introduction to Ergodic Theory,
Springer, 2007 : pp. 75-78, pp. 80-82, pp. 86-91, pp. 95-96, pp. 102-103,
pp. 105.
[3] 李天岩, 數學傳播期刊, 熵(上), Vol.13, No.3, 1989.
[4] Lorenzo Giovanni Bellù, Paolo Liberati, Inequality Analysis The Gini Index,
EASYPol Module 040 : pp. 2-6.
[5] François Bolley. Separability and completeness for the Wasserstein distance.
Séminaire de Probabilités XLI„ Lecture notes in mathematics, 2008, 1934 :
pp.371-377.
[6] 傳染病統計資料查詢系統/ 依傳染病防治法分類/ 第五
類法定傳染病/ 嚴重特殊傳染性肺炎/ 地理分佈, from
https://nidss.cdc.gov.tw/nndss/DiseaseMap?id=19CoV
[7] Department of Household Registration, MOI / 戶政司/ 主題資訊/
人口政策及統計資料/ 人口統計資料庫/ 人口統計資料, from
https://www.ris.gov.tw/app/portal/346