# Concept derivation guides

299 concept teaching guides, separate from the 36,000 individual question derivations.

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Concept guide



## [Covariance: symmetry, the affine rule, and why zero covariance does not imply independence](/en/derivations/derivation-mko-covariance-01/?from=%2Fen%2Fderivations%2F%3Fq%3Dmko-covariance)

On one exact finite joint law: (1) prove Cov(Y,X)=Cov(X,Y); (2) prove Cov(aX+b,cY+d)=ac·Cov(X,Y); (3) prove that independence of X and Y gives Cov(X,Y)=0, so a nonzero covariance refutes independence; (4) for the joint law with X uniform on −1, 0, 1 and Y=X², compute E[X], E[Y], E[XY] and Cov(X,Y)=0, and prove that X and Y are not independent.

Covariance

derivation-mko-covariance-01





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