The Sample and Its Basic Characteristics

Find the distribution function of the $k$-th order statistic $X_{(k)}$, $k = 1, \ldots , n$.

The sample corresponds to the distribution $R[0 ; 1]$. Compute $\mathbb {E}\left[X_{(n)}\right]$ and $\operatorname {Var}\left[X_{(n)}\right]$.

The sample corresponds to a distribution $F(x)$ with finite moment $\nu_{r}$. Prove that $\bar{\mu }_{r}(n) \xrightarrow {\text{ a.s. }} \mu_{r}, n \rightarrow \infty$.

A sample of size $n \gg 1$ corresponds to the distribution $\mathcal{N}\left(0 ; \sigma^{2}\right)$. Find the distribution of the sample moment $\bar{\nu }_{2}(n)$ as $n \rightarrow \infty$.

A two-dimensional sample of size $n \gg 1$ corresponds to the distribution $\mathcal{N}(\mu ; K)$, where the covariance matrix is $K = \left[\begin{array}{cc}D_{X} & k_{X Y} \\ k_{Y X} & D_{Y}\end{array}\right]$. Prove that $\sqrt{n}\left(\widehat{k}_{X Y}(n)-k_{X Y}\right) \xrightarrow {d} \xi \sim \mathcal{N}\left(0 ; D_{X} D_{Y}+k_{X Y}^{2}\right)$ as $n \rightarrow \infty$.

The sample corresponds to the distribution $\operatorname {Exp}\left(\lambda \right), \lambda > 0$. Find the limit to which $\bar{\nu }_{2}(n)$ converges almost surely as $n \rightarrow \infty$.

A sample of size $n \gg 1$ is generated by a random variable $X \sim \operatorname {Exp}\left(1\right)$. Estimate $\mathbb {P}\left(\left|\widehat{F}_{n}(1)-F_{X}(1)\right| \leqslant \frac{1}{\sqrt{n}}\right)$.