Ch2probrev - Introduction to Financial Econometrics Chapter...

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Introduction to Financial Econometrics Chapter 2 Review of Random Variables and Probability Distributions Eric Zivot Department of Economics, University of Washington January 18, 2000 This version: February 21, 2001 1 Random Variables We start with a basic de&nition of a random variable De&nition 1 A Random variable X is a variable that can take on a given set of values, called the sample space and denoted S X , where the likelihood of the values in S X is determined by X & s probability distribution function (pdf). For example, 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 random variable. The price next month must be positive and realistically it can± t get too large. Therefore the sample space is the set of positive real numbers bounded above by some large number. It is an open question as to what is the best characterization of the probability distribution of stock prices. The log-normal distribution is one possibility 1 . As another example, consider a one month investment in Microsoft stock. That is, we buy 1 share of Microsoft stock today and plan to sell it next month. Then the return on this investment is a random variable since we do not know its value today with certainty. In contrast to prices, returns can be positive or negative and are bounded from below by -100%. The normal distribution is often a good approximation to the distribution of simple monthly returns and is a better approximation to the distribution of continuously compounded monthly returns. As a &nal example, consider a variable X de&ned to be equal to one if the monthly price change on Microsoft stock is positive and is equal to zero if the price change 1 If P is a positive random variable such that ln P is normally distributed the P has a log-normal distribution. We will discuss this distribution is later chapters. 1
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is zero or negative. Here the sample space is trivially the set { 0 , 1 } . If it is equally likely that the monthly price change is positive or negative (including zero) then the probability that X =1 or X =0 is 0 . 5 . 1.1 Discrete Random Variables Consider a random variable generically denoted X anditsseto fpo ss ib leva lueso r sample space denoted S X . De&nition 2 A discrete random variable X is one that can take on a &nite number of n di f erent values x 1 ,x 2 ,...,x n or, at most, an in&nite number of di f erent values x 1 2 ,.... De&nition 3 The pdf of a discrete random variable, denoted p ( x ) , is a function such that p ( x )=Pr( X = x ) . The pdf must satisfy (i) p ( x ) 0 for all x S X ; (ii) p ( x )=0 for all x/ S X ; and (iii) P x S X p ( x )=1 . As an example, let X denote the annual return on Microsoft stock over the next year. We might hypothesize that the annual return will be in! uenced by the general state of the economy. Consider &ve possible states of the economy: depression, reces- sion, normal, mild boom and major boom. A stock analyst might forecast di f erent values of the return for each possible state. Hence X is a discrete random variable
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Ch2probrev - Introduction to Financial Econometrics Chapter...

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