- 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).