Suppose we are given $(X_1,...,X_n)$ random variables which are iid. from $\mathcal{N}(\mu,\theta)$ and finite variance. Let $Y=\frac{1}{n}\sum_{i=1}^n(X_i-\overline X)^2$ and define a loss function $L(\theta, d(y))=(\theta-d(y))^2$, where $d(y)=by$ where $b$ is constant.
I want to compute $R(\theta, d)=E_\theta((\theta - by)^2)$.
This is my first exercise in decision theory, so I am not sure what to do next. I assume, that by $y$ we mean a realization of $Y$.
The only thing I can think of is doing something like this:
$R(\theta, d)=E_\theta(\theta^2 -2bY\theta +b^2Y^2)= E_\theta(\theta^2)-2b\theta E_\theta(Y)+b^2E_\theta(Y^2)) $
However, I have no idea on how to compute the actual terms. If we consider $\theta$ as random variable, then what is its expectation?
EDIT: Using that $Y\sim \chi^2(n)\theta$ I obtain
$R(\theta, d)= E_\theta(\theta^2)-2b\theta E_\theta(Y)+b^2E_\theta(Y^2))=\theta^2 -2bn\theta^2 +b^2E_\theta[(\theta\chi^2(n))^2] $
