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Remark 99 If random variables X and Y are independent, then they are unM correlated (but not conversely). Proof. We we will show it only for random variables ...

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Download Properties of covariance and covariance matrix and more Slides Advanced Calculus in PDF only on Docsity! Chapter 9
Properties of covariance
and covariance matrix
9.1 Properties
Remark 97 In order to avoid unnecessary formal complications we will as-
sume that all considered random variables have expected values and if only it is
necessary variances
1
Va,b,c,deR cov(aX +6,cY +d) = accov(X,Y) (9.1)
Since we have: E(aX +6)(cY +d) — E(aX + b)E(cY +d) =
acEXY + adEX + beBY + bd -acEXEY —beEX —adEY —bd
= accov(X,Y)
2
cov(X,Y) =cov(¥ — EX,Y- EY) =E(X-EX)(Y-EY) (92)
It follows immediately from point (9.1) with a =c=1andb =—EX,d
= —EY and the fact that E(X — BX) =0
3
V(X) >0 |
It follows from the formula (9.2) with X =Y
4
cov(X,Y) = cov(Y,X); (9.3)
cov(X,Y+Z) = cov(X,Y) +cov(X,Z)
The proof of these facts is trivial
61
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