Covariance times constant, basic rules
By Jessica Wood •
I know the from the basic rule of the covariance we have:$$\text{Cov(aX,Y)=aCov(X,Y)}$$however now i'm looking at a case that is creating me some doubt: Looking at the covariance of the same random variable:
$1)$ $\text{Cov(aX,X)=aCov(X,X)=aVar(X)}$
$2)$ $\text{Cov(aX,X)=Var(aX)=}a^2\text{Var(X)}$
which one is the correct solution?
Thank you in advance
$\endgroup$ 21 Answer
$\begingroup$The first is correct.
The second is not: By definition $\operatorname{Cov}(X,X)=\operatorname{Var}(X)$.
So $$\operatorname{Var}(aX)= \operatorname{Cov}(aX,aX)= a\operatorname{Cov}(X,aX)= a^2\operatorname{Cov}(X,X)=a^2 \operatorname{Var}(X)$$
So the last $=$ of 2 is correct, the first is not.
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