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# class03 - Random Variables and Probability Distributions...

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Statistics for Business Control and Regression Analysis – Spring 2008 1 Random Variables and Random Variables and Probability Distributions Probability Distributions

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Statistics for Business Control and Regression Analysis – Spring 2008 2 Random Variables Random Variables Question: A coin is tossed three times; how many “heads” appeared? A random variable (RV) is “something that takes on numerical values depending on chance” Usually denoted by the capital letters X, Y, Z Examples Value of a stock next month Time I’ll wait for the subway today Number of iPods to be sold this year
Statistics for Business Control and Regression Analysis – Spring 2008 3 Tossing a Coin Three Times Tossing a Coin Three Times Define: X = number of “heads” Outcome Prob. X H H H 1/8 3 H H T 1/8 2 H T H 1/8 2 H T T 1/8 1 T H H 1/8 2 T H T 1/8 1 T T H 1/8 1 T T T 1/8 0 x P(X = x) 0 1/8 1 3/8 2 3/8 3 1/8

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Statistics for Business Control and Regression Analysis – Spring 2008 4 Discrete vs. Continuous RVs Discrete vs. Continuous RVs Discrete RV: No “in between” values; often “counting” Number of “heads” Number of iPods sold Sum of two dice rolls Continuous RV: Always “in between” values; often “measuring” Waiting time for subway Tomorrow’s temperature Value of a stock next month
Statistics for Business Control and Regression Analysis – Spring 2008 5 The Probability Distribution Function The Probability Distribution Function When X is a discrete RV, we define p(x) = P(X = x) p(x) is the probability distribution function (PDF) of X In the three-tosses example: p(2) = 3/8 Properties of the probability distribution function 0 ≤ p(x) ≤ 1 p(x) = 1 P(X = a or X = b) = p(a) + p(b) x all

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Statistics for Business Control and Regression Analysis – Spring 2008 6 The Probability Distribution Function The Probability Distribution Function Graphically: Total shaded area is 1 3 2 1 0 0.4 0.3 0.2 0.1 0.0 p(x) x
Statistics for Business Control and Regression Analysis – Spring 2008 7 The Cumulative Distribution Function The Cumulative Distribution Function The cumulative distribution function (CDF) of X is F(x) = P(X ≤ x) In the three-tosses example, F(2) = P(X ≤ 2) = P(X = 0 or X = 1 or X = 2) = p(0) + p(1) + p(2) = 1/8 + 3/8 + 3/8 = 7/8

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class03 - Random Variables and Probability Distributions...

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