How to compute correlation from variances

Q: How to compute correlation from variances

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Question

You have two random variables $X$ and $Y$ .

You are told:

$\mathrm{Var}(X+Y)$
$\mathrm{Var}(X)$

Is this information sufficient to determine the correlation $\rho_{X,Y}$ ? If not, what additional quantity(ies) would you need?
If you also know $\mathrm{Var}(Y)$ , derive $\rho_{X,Y}$ in terms of $\mathrm{Var}(X+Y)$ , $\mathrm{Var}(X)$ , and $\mathrm{Var}(Y)$ .
Special case: if you assume $\mathrm{Var}(Y)=\mathrm{Var}(X)$ , simplify the expression for $\rho_{X,Y}$ .

How to compute correlation from variances

Solution

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