Can anybody help me to solve this question? Thanks in advance,
Let $X_1,...,X_n \sim N(\mu,\sigma^2); (\mu,\sigma^2) \in R\times R_+, n\geq 2$ and $Y=(\frac{1}{n}\Sigma_{i=1}^n X_i,\frac{1}{n}\Sigma_{i=1}^n X_i^2)$
Prove that this yields an $SEF_2$ with canonical parameter $(\nu_1,\nu_2)=(\frac{n\nu}{\sigma^2},-\frac{n}{2\sigma^2})$ and canonical statistic Y, Furthermore, show that the dominating measure is:
$$\nu(dy)=(\frac{n}{2})^{\frac{n}{2}}(\sqrt{\pi}\Gamma (\frac{n-1}{2}))^{-1}(y_2-y_1^2)^{\frac{n-3}{2}}\lambda(dy_1,dy_2)$$
$\kappa = {(y_1,y_2):y_1^2\leq y_2}$ and 0 otherwise.