Recent content by baiyang11

I appreciate your continuous attention on this problem and I do hope attentions from some other members, maybe from various areas, because it seems to be interdisciplinary between set theory and probability theory. This abstracted problem comes from a concrete problem in my major, which is...

Thank you for reply. You provide a clear example when $A$ and $B$ have nothing in common with $I_{1}$ at all. But I think my original intention is to make $|A \cap I_{1}| \geq |B \cap I_{1}|>0$. I admit it is my mistake not to make this clear. So if $|A \cap I_{1}| \geq |B \cap I_{1}|>0$, is...

Consider N random variables X_{n} each following a Bernoulli distribution B(r_{n}) with 1 \geq r_{1} \geq r_{2} \geq ... \geq r_{N} \geq 0. If we make following assumptions of sets A and B: (1) A \subset I and B \subset I with I=\{1,2,3,...,N\} (2) |A \cap I_{1}| \geq |B \cap I_{1}| with...

Thank you for reply! (1) Yes. Here the 'E' character means expected value. I just didn't know how to make it like your character when I wrote this question. Now I know, but I can't edit it. (2) Yes. $A \subset I$ and $B \subset I$. Since I can't edit the original post, let me repost the question...

Dear all, I have a question attached related to both probability and cardinality. Let me know if my formulation of the problem is non-rigorous or confusing. Any proof or suggestions are appreciated.Thank you all. The question follows.Consider a set \(I\) consists of \(N\) incidents...

(1) Given a twice continuously differentiable function f(x),x\in\mathbb{R}, it can be justified that f''(x) is not always positive for \forall x\in\mathbb{R}. However, if f''(x_0)>0, is f(x) ("locally") convex in some epsilon distance around x_0? (As shown in the 1st picutre in #1) (2) Given a...

Thank you! I've edited the #1 and ask the question in text form.

Convex function and convex set(#1 edited) Please answer #4, where I put my questions more specific. Thank you very much! The question is about convex function and convex set. Considering a constrained nonlinear programming (NLP) problem \[min \quad f({\bf x}) \quad {\bf x}\in \mathbb{R}^{n}...