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Statistics 270 - Lecture 20
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Last Day…completed 5.1 Today Section 5.2 Next Day: Parts of Section 5.3 and 5.4
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Expected Value for Joint Distributions If X and Y are jointly distributed rv’s, then the expected value of a function, h(X,Y), is given by Continuous Discrete
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Example An instructor has given a short test consisting of two parts For a randomly selected student, let X = # points earned on the first part and Y =#points earned on the second part Suppose that the joint pmf of X and Y is given by
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Example If the score recorded in the grade book is the total number of points earned on the two parts, what is the expected recorded score?
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Show that if X and Y are independent random variables, then E(XY)=E(X)E(Y)
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Covariance Suppose X and Y are jointly distributed rv’s If X and Y are dependent, one is frequently interested in the strength of their relationship This can be measured by the covariance Gives both direction and magnitude and direction of association
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Covariance If X and Y are jointly distributed rv’s, then the covariance between X and Y is Continuous Discrete
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Covariance Visually
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Covariance – Short-cut If X and Y are jointly distributed rv’s, then the covariance between X and Y is Cov(X,Y)=E(XY)-E(X)E(Y)
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Example An instructor has given a short test consisting of two parts For a randomly selected student, let X = # points earned on the first part and Y =#points earned on the second part Suppose that the joint pmf of X and Y is given by
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Example Find the covariance between X and Y
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Correlation Covariance has the drawback that the value depends on the units of measurement…hard to interpret A related, unit free, measure is the correlation coefficient
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Correlation Properties:
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Example Find the correlation between X and Y in the previous example
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