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If I have two random variables $X$ and $Y$ and their variance is the same, i.e., $Var(X)=Var(Y)$. Does this mean $Var(X)=Var(Y)=Cov(X,Y)$ ?

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    $\begingroup$ Roll two fair dice one on Tuesday and the other on Saturday. Their distributions have the same spread (and the same variance), but their covariance should presumably be zero. $\endgroup$
    – Glen_b
    Commented Apr 20, 2022 at 4:40

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In short, no.


Consider the counterexample that if two normal variables both have unit variance, their covariance could be zero.


Alternatively, for geometric motivation, consider that two vectors can have the same length while pointing in perpendicular/orthogonal directions.

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