query: Prove: $m$ is a median of $X$ iff the function $g(a)=\mathbb E(|X-a|)$ is minimized at $a=m$
positive_passages.docid: 25605
positive_passages.text: Why does the median minimize $E(|X-c|)$?
negative_passages.docid: 77192
negative_passages.text: How to prove $\text{Rank}(AB)\leq \min(\text{Rank}(A), \text{Rank}(B))$?
negative_passages.docid: 105268
negative_passages.text: prove inequality through mean value theorem
negative_passages.docid: 34325
negative_passages.text: Minimizing $\sum_{i=1}^n \max(|x_i - x|, |y_i - y|)$ - Sum of Max of Absolute Values
negative_passages.docid: 6410
negative_passages.text: Prove that $(ma, mb) = |m|(a, b)\ $ [GCD & LCM Distributive Law]
negative_passages.docid: 36099
negative_passages.text: Show that $\|f\|^{2}$ attains a minimum value on the interior of $B$
negative_passages.docid: 24988
negative_passages.text: Prove there exists $m > 2010$ such that $f(m)$ is not prime
negative_passages.docid: 93684
negative_passages.text: Show that $\mu (E \setminus A)>0$ if $A$ is a nonmeasurable subset of $X$ and $A \subseteq E$, $E$ is measurable.
negative_passages.docid: 46056
negative_passages.text: If $\mathcal{E}$ are subsets of $X$ and $A \subset X$. Show that $\mathcal{A}\mathcal{(E \cap}$ A)=$\mathcal{A}\mathcal{(E)} \cap$ A
negative_passages.docid: 42366
negative_passages.text: Prove $\mathbb{P}(\sup_{t \geq 0} M_t > x \mid \mathcal{F}_0)= 1 \wedge \frac{M_0}{x}$ for a martingale $(M_t)_{t \geq 0}$
negative_passages.docid: 119735
negative_passages.text: Show that $\mathbb{E}(g(T-p)) < \mathbb{E}(g(S-p))$ for any convex function $g$ if $T$ and $S$ are estimators of $p$
negative_passages.docid: 123753
negative_passages.text: In a metric $(X,d)$, prove that for each subset $A$, $x\in\bar{A}$ if and only if $d(x,A)=0.$
negative_passages.docid: 27170
negative_passages.text: $a\in \mathbb{N}$, $p$ prime, $a<p$ prove that $a\mid p+1\iff\exists\, b,c\in\Bbb N:\dfrac{a}{p}=\dfrac{1}{b}+\dfrac{1}{c}$
negative_passages.docid: 62270
negative_passages.text: Given two increasing continuous functions $f,g$ prove that $(b-a) \int^b_a f(x)g(x) dx > \int^b_a f(x) dx \int^b_a g(x) dx$
negative_passages.docid: 144745
negative_passages.text: Proof of: If $x_0\in \mathbb R^n$ is a point of local minimum of $f$, then $\nabla f(x_0) = 0$.
negative_passages.docid: 78471
negative_passages.text: Closure of $\{\frac{1}{n}\}:=M$
negative_passages.docid: 154247
negative_passages.text: Proving $f(x)$ attains $\max$ or $\min$ when $f(x)\to0$ as $|x|\to\infty$.