Let $X_1$,$X_2$...be i.i.d. with mean 1 and variance 1. Let $\epsilon_1$,$\epsilon_2$ be i.i.d with mean 0 and variance $\sigma^2$, independent of the $X_i$. Let $Y_i$=$\theta$$X_i$+$\epsilon_i$ for all i. $X_i$ and $Y_i$ are observed, but $\epsilon_i$ are unobservable and $\theta$ is unknown.
What is the covariance of ($X_i$,$Y_i$)?
So $E(Y_i)=E[\theta X_i+\epsilon_i]=\theta$
$Var(Y_i)=Var[\theta X_i+\epsilon_i]=\theta^2 +\sigma^2$
Then I use the Covariance formula and lost at how to compute $Cov=E(X_i *(\theta$$X_i+\epsilon_i))-E(X_i)E(Y_i)$