Let $(X,Y)\sim N_2(\mu,\Sigma)$ with $ \mu =(\mu_x,\mu_y)$. Also,let $\sigma_x^2,\sigma_y^2$ and $\sigma_{xy} $ denote the variance of $X$, variance of $Y$ and covariance between $X$ and $Y$ , respectively. Let $E[Y|X] = \beta_0 + \beta_1 X$. (a) Consider taking $n$ independent observations from the joint normal distribution and let $\hat{\beta_1 } $ be the MLE of $\beta_1$. Find the variance of $\hat{\beta_1}$. What happens to this variance as $n \rightarrow \infty$? (b) Again consider a random sample of size n from the joint distribution of $(X, Y )$. Let $\hat{\sigma^2}$ denote the (bias corrected) MLE of $\sigma^2$ . Find the variance of $\hat{\sigma^2}$ and use this to show that $\hat{\sigma^2}$is a consistent estimator of σ .
I know that $\hat{\beta_1} = \frac{S_{xy}}{S_{xx}}$, $Var(\hat{\beta_1}) = \frac{\sigma^2}{\sum_i x_i^2 - n\bar{x}^2}$ and $\hat{\sigma^2} =s^2$, but I am not sure how to use this info to answer the questions.