lecturenotesweek13

# lecturenotesweek13 - ISyE 2027 Probability with...

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ISyE 2027 Probability with Applications Polly B. He Fall10, Week 13 Reading: Section 10.3, Notes The Correlation Coeﬃcient Last week, we talked about using Covariance to measure the relation between two random variables. What would happen if we change the scale of the r.v.’s? Covariance under change of unit: Let X and Y be two random variables, then Cov( rX + s, tY + u ) = rt Cov( X, Y ) for all numbers r, s, t and u . This rule basically says that the covariance changes when we change the scale of the r.v.’s, which is a “bad” (counter-intuitive) measure of the relation between r.v.’s. For example, the relation of a person’s weight and height doesn’t change when we switch from measuring heights in inches to centimeters - the shape of the person is still the same! Deﬁnition: Let X and Y be two random variables. The correlation coeﬃcient ρ ( X, Y ) is deﬁned to be 0 if Var( X ) = 0 or Var( Y ) = 0, and otherwise ρ ( X, Y ) = Cov( X, Y ) Var( X )Var( Y ) Note: 1. Correlations ρ ( X, Y ) are between -1 and 1 (proof in textbook, p. 143.) Recall that probabilities are between 0 and 1.

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lecturenotesweek13 - ISyE 2027 Probability with...

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