Let X1,...,Xn denote a random sample from a $\displaystyle N(\mu , \sigma)$ distribution. Let $\displaystyle Y = \Sigma \frac{(X_i - \overline{X})^2}{n}$and let $\displaystyle L(\theta , \delta (y)) = [\theta - \delta(y)]^2$ and $\displaystyle \delta (y) = by$ where b does not depend on y.

Show that $\displaystyle R(\theta^2, \delta) = \frac{\theta^2}{n^2}[(n^2 - 1)b^2 -2n(n-1)b + n^2]$

So:

$\displaystyle R(\theta^2, \delta) = E(L(\theta , \delta (y))) = E([\theta - \delta(y)]^2) = E((\theta -by)^2) = E(\theta^2 -2 \theta by +b^2y^2)$

Kinda stuck here. Not sure what the expectation of $\displaystyle Y = \Sigma \frac{(X_i - \overline{X})^2}{n}$ is.

Any help would be appreciated.