Knowing mean and need to find covariance

Question

I've proven that $E(U)=0$, and now to find $Cov(U,X)$, according to the formula: $Cov(U,X)=E[U-E(U)][X-E(X)]$, we can't find the covariance. So we need to use another formula: $Cov(U,X)= [Var(U+X)-Var(U)-Var(X)]/2$. But how to find Cov(U+X)? They are not independent, so it is not $Cov(U)+Cov(x)$. Thanks.

Answer 1

$E[U] = $$E[Y] + \beta E[X] - \alpha\\ \mu_Y + \beta \mu_X - \alpha$

$\alpha = \mu_Y + \beta \mu_X$

$E[U] = 0$

That was the easy one.

We can break $Y$ into components that are dependent and independent from $X.$ $Y = Z + \frac {Cov(X,Y)}{\sigma_x^2} X = Z + \beta X$

$U = Z + \beta X -\alpha - \beta X = Z-\alpha$

$Cov (Z,X) = 0$

$Cov (U,X) = 0$

Answer 2

Hint: $\mathsf {Var}(Y)=\mathsf {Var}(\alpha+\beta X+U)$

Answer 3

Why can't we just use the first Formula?

$Cov(U,X)=E[[U-E(U)][X-E(X)]]=E[U\cdot X-U \cdot E[X]]=E[UX]-E[X]E[U]=E[UX]=E[XY] -\alpha E[ X]-\beta E[X^{2}]$

E[X] is known, $E[X^{2}]$ encoded in the Variance of X and E[XY] in the covariance of X and Y which can be obtained from the correlation coefficent.