Mathematica 這套軟體是一個極具能力又具伸縮性的符號計算代數系統。
它可以幫助使用者處理複雜的代數工作，也可以輕易的應付數值和圖形的部分。動差生成函數和特徵函數的推導是其中一項工作，在辨別分佈上它明顯是有效的工具。在這篇論文中，我們在Mathematic 內建立一些規則來幫助我們計算獨立隨機變數線性組合後的動差生成函數和特徵函數。這些指令利用式樣配合的規則來增加 Mathematic 的能力來簡化包含代數項乘積的式子。這增加 Mathematic 的函數能力對於動差生成函數和特徵函數可以是特別有益處的。這些規則的運用來決定平均數、變異數和各種隨機變數的分佈可以被說明。

Mathematica is an extremely powerful and flexible symbolic
computer algebra system that enables the user to deal with
complicated algebraic tasks. It can also easily handle the
numerical and graphical sides. One such task is the derivation of
moment generating functions (MGF) and characteristic functions
(CF), demonstrably effective tools to characterize a distribution.
In this paper, we define some rules in Mathematica to help in
computing the MGF and CF for linear combination of independent
random variables. These commands utilizes pattern-matching code
that enhances Mathematica's ability to simplify expressions
involving the product of algebraic terms. This enhancement to
Mathematica's functionality can be of particular benefit for MGF
and CF. Applications of these rules to determine mean, variance
and distribution are illustrated for various independent random
variables.

Introduction …………………………………………………………………………………1
Extending Mathematica………………………………………………………………………1
Applications of moment generating and characteristic functions………………3
Summary…………………………………………………………………………………………8
References……………………………………………………………………………………9
Appendix………………………………………………………………………………………10

1. Feller, W. (1966).
An Introduction to Probability Theory and its Applications.
Volume II, New York: John Wiley.

2. Grimmett, G. and Welsh, D. (1986).
Probability: An Introduction}. Oxford, U.K.: Oxford University Press.

3. Wolfram, S. (1999).
The Mathematica Book}.
Wolfram Media/Cambridge University Press, 4th ed.