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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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