Calculation of correlation with knowledge of expected values and variances

statistics covariance correlation variance

My book problem states that U and V are independent variables, each with expected value $\mu$ and variance $\sigma^2$. We then have Z = $\alpha U + V\sqrt{1-\alpha^2}$, and I am asked to calculate the correlation coefficient for U and Z, $\rho_{uz}$. I know correlation is derived with $\frac{cov(U,Z)}{\sqrt{var(U)var(Z)}}$, and I'm able to calculate var(U) and var(Z) but not cov(U,Z). I'm aware Cov(U,Z) = E(UZ) - E(U)E(Z), and again I have E(U) and E(Z), but I have no way of getting E(UZ) that I'm aware of...what am I missing?


$cov(U,Z) = cov(U,\alpha U+V\sqrt{1-\alpha^2})$

$=cov(U,\alpha U)+cov(U,V\sqrt{1-\alpha^2})$

$=\alpha var(U)$

The second line and third lines are standard properties of covariance that are useful for problems like this (not hard to show they're true from the definitions).

That's all you need to solve the problem if you know how to get the variances.

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