(a) Derive the log-likelihood of the parameters for these data.
(b) Derive the maximum likelihood estimates for the mean $\mu$ and variance $\Sigma$ by setting the derivative of the log-likelihood to zero.
[2] (#) Shortly explain why the Gaussian distribution is often preferred as a prior distribution over other distributions with the same support?
[3] (###) Proof that the Gaussian distribution is the maximum entropy distribution over the reals with specified mean and variance.
[4] (##) Proof that a linear transformation $z=Ax+b$ of a Gaussian variable $\mathcal{N}(x|\mu,\Sigma)$ is Gaussian distributed as
$x \sim \mathcal{N}(\mu_x,\sigma_x^2)$ and $y \sim \mathcal{N}(\mu_y,\sigma_y^2)$, what is the PDF for $z = A\cdot(x -y) + b$?