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# Lect8 - Random Variables Properties of Mean Variance and...

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Random Variables r Outline Random Variables Mean, Variance and Standard Deviation Properties of Mean, Variance and Standard Deviation Comparing Random Variables Portfolios and Random Variables 1 / 18 ISOM 2500 Lect 8: Random Variables, and Joint Probability Distributions

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Random Variables r Mean, Variance and Standard Deviation Random Variables Q1 : If a day trader buys a certain stock and holds it for one day, what will be her/his return? Q2 : In the Big and Small game, if a player bets \$1 on “Big”, how much will s/he earn or lose? In answering these questions, one uses random variables , which describe the uncertain numerical outcomes of a random process; often denoted by a capital letter like X , Y , Z , etc. Two Types: Discrete vs. Continuous Discrete : A random variable that takes on one of a list of possible values Continuous : A random variable that takes on any value in an interval 2 / 18 ISOM 2500 Lect 8: Random Variables, and Joint Probability Distributions
Random Variables r Mean, Variance and Standard Deviation Stock Return Example Let X be the return of holding the stock for one day. Stock returns usually take many values (recall the histogram of HSBC returns in Lect 3). For illustration purpose, let’s assume that for this stock, X takes value - 5 %(= - 0 . 05 ) , 0 and 5 %(= 0 . 05 ) , with respective probabilities 0 . 09 , 0 . 8 and 0 . 11. To summarize such an X , we can use either a table x - 0 . 05 0 0.05 p ( x ) = P [ X = x ] 0.09 0.8 0.11 (the function p ( · ) is called the probability distribution function or probability mass function ) or a plot of the probability distribution function: 3 / 18 ISOM 2500 Lect 8: Random Variables, and Joint Probability Distributions

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Random Variables r Mean, Variance and Standard Deviation Play Big and Small (Sic Bo) 3 fair dice, a player can bet on “Big” (win if the total score will be between 11 and 17 (inclusive) with the exception of a triple) “Small” (win if the total score will be between 4 and 10 (inclusive) with the exception of a triple) ... For \$1, place a bet on “Big”, the house pays you \$1 if it turns out to be “Big” Let X be your net winnings. What are the possible values that X takes? ; respective probabilities: 4 / 18 ISOM 2500 Lect 8: Random Variables, and Joint Probability Distributions
Random Variables r Mean, Variance and Standard Deviation Mean of a Random Variable Suppose a random variable X takes values x 1 , x 2 ,..., x k , with respective probabilities p ( x 1 ) , p ( x 2 ) p ( x k ) Then the mean or expected value of X , denoted by E ( X ) or the Greek letter μ

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Lect8 - Random Variables Properties of Mean Variance and...

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