本論文著重於 Mathematica 7.0 (Wolfram, 2008) 在分配理論上符號運算的應用，研究分成二個部份，首先，我們將會藉由修改一些程式去擴展 Mathematica 的能力以便於去處理獨立單變量隨機變數線性組合的特徵函數之符號運算，這些程式使用 pattern-matching 的語法去提高 Mathematica 在化簡代數乘積及加總的能力。第二，通過 Mathematica 的 pattern-matching 特性，特徵函數可
以被分類到一些常見的分配裡，包括六個離散分配及七個連續分配。最後，將會提供幾個例子，這些例子包含獨立隨機變數線性組合的特徵函數之極限、編寫的程式之應用以及舉例介紹中央極限定理、大數法則及一些分配的性質。

This paper deals with the applications of symbolic computation of Mathematica 7.0 (Wolfram, 2008) in distribution theory. The purpose of this study is twofold. Firstly, we will implement some functions to extend Mathematica capabilities to handle symbolic computations of the characteristic function for linear combination of independent univariate random variables. These functions utilizes pattern-matching codes that enhance Mathematica's ability to simplify expressions involving the product and summation of algebraic terms. Secondly, characteristic function can be classified into commonly used distributions, including six discrete distributions and seven continuous distributions, via the pattern-matching feature of Mathematica. Finally, several examples will be presented. The examples include calculating limit of characteristic function of linear combinations of independent random variables, and applications of coded functions and illustrate the central limit theorem, the law of large numbers and properties of some distributions.

Contents
1 Introduction 3
2 Extending Mathematica 5
2.1 Modifying built-in function . . . . . . . . . . 5
2.2 Match distribution by characteristic function . . . . . 10
3 Examples 14
4 Summary 18
Appendix 19
A.1 Table of characteristic functions of common distributions . . . . . . . . . . 19
A.2 Programs for matching distributions . . . . . . . . . . . 22

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