Top network posts

• 11 If $\mathcal{B}$ is a base of a topology space $\left(X,\tau\right)$, then the Borel $\sigma$-algebra is generated by $\mathcal{B}$?
• 11 Are all bilinear forms identically zero?
• 11 Suppose that for each $a\in \mathbb{C}$ at least one coefficient of the Taylor's series $f$ about $a$ is zero. Show that $f$ is a polynomial.
• 9 There exists polynomial $(P_{n})_{n\in\mathbb{N}}$ such that $P_n(0)=1$, and $\lim_{n\rightarrow \infty}P_{n}(z)=0$..
• 9 Let $X_{(1)}\leq X_{(2)}$ be the order statistics. Evaluate $\operatorname{Var}(X_{(j)})$, $\operatorname{Cov}(X_{(1)},X_{(2)})$
• 9 If $X\sim \mathrm{lognormal}$ then $Y:=(X-d|x\geq d)$ has approximately a Generalized Pareto distribution.
• 7 Define $\rho(f,g):=\int \frac{|f-g|}{1+|f+g|}d\mu.$. Show that $f_{n}\rightarrow f$ in measure $\Longrightarrow$ $\rho(f_{n},f)\rightarrow 0$.
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