@MISC{11chapter1, author = {}, title = {Chapter 1 Probability Concepts}, year = {2011} }

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Abstract

This chapter reviews basic probability concepts that are necessary for the modeling and statistical analysis of financial data. 1.1 Random Variables We start with the basic definition of a random variable: Definition 1 A Random variable X is a variable that can take on a given set of values, called the sample space and denoted SX, where the likelihood of the values in SX is determined by X’s probability distribution function (pdf). Example 2 Future price of Microsoft stock Consider the price of Microsoft stock next month. Since the price of Microsoft stock next month is not known with certainty today, we can consider it a randomvariable. Thepricenextmonthmustbepositiveandrealistically it can’t get too large. Therefore the sample space is the set of positive real numbers bounded above by some large number: SP = {P: P ∈ [0,M], M>0}. Itisanopenquestionastowhatisthebestcharacterizationofthe probability distribution of stock prices. The log-normal distribution is one possibility1. ¥ 1If P is a positive random variable such that ln P is normally distributed then P has a log-normal distribution.