I have a random variable $x = \theta + \epsilon$ where $\theta \sim N(\mu,\alpha^{-1})$ and $\epsilon \sim N(0,\beta^{-1})$, with $Cov(\theta, \epsilon)=0$.
I want to calculate $Cov(x,\theta)$.
What I did is ($E$ denotes expected value function):
$Cov(x,\theta) = E(x\theta)-E(x)E(\theta) = E(\theta^{2}+\theta\epsilon)-\mu*\mu = \mu^{2}-0-\mu^{2} = 0$.
But I am fairly sure it is wrong. Could anyone kindly work this out?
I need this covariance to calculate the expected value and variance of the conditional process $x|\theta$.
Many thanks in advance.
$$Cov(x,θ)=E(xθ)−E(x)E(θ)$$ $$=E(θ^2+θϵ)−E(x)E(θ)$$ $$=E(θ^2)+E(θ)E(ϵ)−E(x)E(θ)$$ $$=(α^{-2}+μ^2)+μ0−μ^2$$ $$=α^{-2}$$