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Suppose $f(x,y)$ is bivariate normal. You are given $E(X)$, $E(Y)$, $\text{Var}(X)$ and $\text{Var}(Y)$. You are also given $\rho$ the correlation coefficient. How would one find $\text{Var}(Y|X=k)$? Would you use the total law of variance? We know that $\rho = \frac{\text{Cov}(X,Y)}{\sigma_x \sigma_y}$.

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Check the Wikipedia article.

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