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.