Scratch pad → MIT 18.6501x Fundamentals of Statistics

If $X_1, \dots, X_n$ are iid from a uniform distribution, $\mathrm{Unif}(0, \theta)$, then $Z = \max \left\{ X_1, \dots, X_n \right\}$ has the cumulative distribution given by
$$F_Z(z) = \frac{z^n}{\theta^n}.$$
Definition. An estimator $\theta_n$ is consistent if it converges in probability to the true value of the parameter $\theta^\star$, that is, $\theta_n \xrightarrow[(p)]{n \to \infty} \theta^\star$.
Definition. A random vector is a Gaussian vector if any linear combantion of its components is a (univariate) Gaussian random variable; that is, $\mathbf\alpha^\top\mathbf{x}$ is Gaussian for any non-zero vector $\mathbf\alpha$. I'm surprised I was not aware of this definition.
An alternate formula for the covariance is $\mathrm{cov}(X, Y) = \mathbf{E}\left[X(Y - \mu_Y)\right]$; this property can be used to prove that the covariance is bilinear:
$$\mathrm{cov}(aX + bY, Z) = a \, \mathrm{cov}(X, Z) + b \, \mathrm{cov}(Y, Z). $$
If $(X, Y)$ is independent of $(U, V)$ then all the following pairs are independent: $(X, U)$, $(X, V)$, $(Y, U)$, $(Y, V)$.
Question. What can we estimate in statistics? It seems that is not only about the parameters, but also other functions related to the true probability distribution (such as, expectation or covariance).