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If $A$ is an independent $RV$. Then we know that
Var$[A+A] = $Var$[A] + $Var$[A]$

This also means that Cov$[A,A] = 0$ and likewise Cov$[A,A] = $Var$[A] = 0$.

So Var$[A+A] = 0 + 0 + 0$.

This does not seem right to me, can someone point out my flaw in logic.

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By saying $A$ is an independent random variable if you mean that $A$ is independent of itself, then all the equalities you have written are correct. In this case $A$ is necessarily a constant random variable so its variance and covariance with itself are both $0$.

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  • $\begingroup$ Thanks makes sense $\endgroup$ – arm.u Feb 14 at 6:16

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