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Unformatted text preview: q q q Statistical methods provide a way of parsimoniously summarizing the strength, pattern, and signiﬁcance of relationships. q q q q q q q q 0 3 6 9 Age 12 15 Researchers are often interested in complex relationships among variables. 18 Figure: Nearest Neighbor Prediction of Age and Lung Capacity for Smokers and Non-Smokers. Culpepper, SA STAT 400: Statistics and Probability I Deﬁnitions Examples Predictions, Best Fit Line 4.1: Distributions of two random variables 4.2: The Correlation Coeﬃcient Deﬁnition of Covariance Researchers are interested in how variables relate. e.g., how do diﬀerences on one variable relate to diﬀerences on another (i.e., how do they covary). Let u (X , Y ) = (X − µX ) (Y − µY ). Note E [u (X , Y )] = E [(X − µX ) (Y − µY )] = E (XY ) − µX µY . In the discrete case, the covariance is deﬁned as, xyf (x , y ) − µX µY σ X ,Y = x (11) y In the continuous case, σX ,Y is, ∞ ∞ −∞ −∞ xyf (x , y ) dxdy − µX µY σ X ,Y = The covariance...
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## This note was uploaded on 12/12/2013 for the course STAT 400 taught by Professor Kim during the Fall '08 term at University of Illinois, Urbana Champaign.

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