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provide a way of
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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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