# What is x y stat 400 statistics and probability i 41

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Unformatted text preview: ﬃcient Deﬁnitions Examples Predictions, Best Fit Line Discrete Covariance Example Suppose f (x , y ) is deﬁned as, y 0 1 2 1 0.15 0.10 0.00 x 2 0.25 0.30 0.20 Culpepper, SA What is E (XY )? What is σX ,Y ? What is ρX ,Y ? STAT 400: Statistics and Probability I 4.1: Distributions of two random variables 4.2: The Correlation Coeﬃcient Deﬁnitions Examples Predictions, Best Fit Line Continuous, Covariance and Correlation Examples Use the deﬁnition of the covariance to prove that, 2 2 Cov (aX + bY , cX + dY ) = ac σX + (ad + bc ) σX ,Y + bd σY Note that Var (aX + bY ) = Cov (aX + bY , aX + bY ) Find ρX ,Y for the following distributions, f (x , y ) = f (x , y ) = 60x 2 y 0 ≤ x ≤ 1, 0 ≤ y ≤ 1, x + y ≤ 1 0 otherwise (14) 12x (1 − x ) e −2y 0 ≤ x ≤ 1, y > 0 0 otherwise (15) f (x , y ) = x +y 3 0 < x < 2, 0 < y < 1 0 otherwise Culpepper, SA STAT 400: Statistics and Probability I (16) 4.1: Distributions of two random variables 4.2: The Correlation Coeﬃcient Deﬁnitions Examples Predictions, Best Fit Line Predictions and The Best Fit Line Sometimes we are interested in predicting Y with values of X . One way of ﬁnding a line of best ﬁt is to minimize the expected squared errors, K (b) = E ((Y − µY ) − b (X − µX ))2 = E (Y − µY )2 − 2b (X − µX ) (Y − µY ) + b2 (X − µX )2 2 2 = σY − 2bρX ,Y σX σY + b2 σX We want to choose b so as to minimize K (b). K (b) = 0 and K (b) > 0 implies the value of b is, σY b = ρX ,Y σX The best ﬁt line is y = µY + b (x − µX ) and 2 K (b) = σY 1 − ρ2 ,Y X Culpepper, SA STAT 400: Statistics and Probability I (17) (18)...
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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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