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Aug
13
comment Interpretation of $\sigma$-algebra and filtrations (follow-up question)
Thank you for answering an older question, it's always a bit frustrating to get no answers. Since posting this question, my understanding of probability has (luckily) expanded, so this all makes much more sense now. And yeah, the non-constructiveness of the conditional expectations keeps bugging me again and again, looking forward to the day when I'll be able to say that I "got it".
Aug
7
comment Why does $[X\in A]=[(X, Y)\in A\times \mathbb R]$
@Berci It appears that other answers are not consistent with your comment (see discussion under Augustin's answer), would you like to elaborate or change it?
Aug
7
comment Why does $[X\in A]=[(X, Y)\in A\times \mathbb R]$
I suppose my fault was thinking that $[(X,Y)\in A\times B] = [X \in A]\times [Y \in B]$, but then was Berci (in the comments) wrong in saying that $[(X,Y)\in A\times\mathbb R] = [X\in A] \times \Omega$ ?
Aug
7
comment Why does $[X\in A]=[(X, Y)\in A\times \mathbb R]$
What if we were interested in $[(X,Y)\in A\times B]$? Since you use the same $\omega$ for both $X$ and $Y$ at the same time, wouldn't that be a problem? Or is that again a case where this notation wouldn't work?
Aug
7
comment Why does $[X\in A]=[(X, Y)\in A\times \mathbb R]$
Thanks (to Stefan Hansen, too), that does make some sense to me, but then, say, what if $Y$ was from a different $\Omega'$? What would the set then look like?
Aug
7
comment Why does $[X\in A]=[(X, Y)\in A\times \mathbb R]$
I agree and understand, but I do not feel this addresses my problem completely. I've added an edit to the question, which might hopefully be steering my thinking into the right direction.
Aug
7
comment Why does $[X\in A]=[(X, Y)\in A\times \mathbb R]$
@Berci: true, an actual question on a Q&A site would probably be appropriate. Added, thanks.
Aug
7
comment Types of Mathematical “Sameness”
You are absolutely correct, @m_squared but making sense of (the most notable) examples, or of some kind of hierarchies can be very useful. I tried my best to make this less of a "give me examples of mathematics" question and more a "let's find the pattern that somehow unifies and categorizes the notion of sameness" question. How much I succeeded, of course, is a different story.
Aug
7
comment Types of Mathematical “Sameness”
@Semiclassical thank you, I was so sure something like this had to already exist!
Aug
6
comment What does it mean to integrate with respect to the distribution function?
Thanks, ah, so is it simply a notation issue and $\int d\mu = \int \mu(dx)$ by definition?
Aug
6
comment What does it mean to integrate with respect to the distribution function?
This is an older answer, but I'll try nonetheless: how is one to understand $\int_\mathbb{R} x \mu_{F}(dx)$? I've never seen that notation and am unsure of what it's supposed to mean.
Jun
7
comment Interval of convergence? (Relatively simple question)
In general, $|x| < 1$ means $x\in (-1,1)$
May
28
comment Strings in a dictionary. A partial order, strict order, and total order?
Hint: What is the relation between the words "cat" and "dog"?
May
11
comment Machine Learning and Probability/Stochastics
The lack of activity on this question, however, does not :(
May
7
comment Machine Learning and Probability/Stochastics
@user237393 thank you! This looks really promising! Makes me quite hopeful in joining these two.
Apr
12
comment Limit w/o L'hopital
@black multiply the fraction by $$ \frac{\sqrt x + \sqrt c}{\sqrt x + \sqrt c} $$
Apr
7
comment $P^n$ transition matrix of a Markov chain
Not sure what you're asking right now. Maybe this will help: $P^n$ actually is the $n$-th power of the transition matrix $P$, only I am trying to get to it through a combinatorial argument, rather than attempting to multiply the matrix $n$-times.
Apr
7
comment $P^n$ transition matrix of a Markov chain
If I stay in state $2$, it means I put a ball inside one of the two boxes which were already nonempty, thus I changed nothing about the number of nonempty boxes and the transition to state $3$ still has the same probability. (Also: It is a homogenous Markov chain)
Apr
7
comment $P^n$ transition matrix of a Markov chain
To add to how $P$ was constructed: the first row, for example, says that with probability $1$ we'll go from state zero to state $1$ - that is, the ball has to be put somewhere, thus making one box nonempty. The second row says that we might hit the one nonempty box with probability $\frac 1k$ or an empty one with probability $\frac {k-1}{k}$
Apr
7
comment $P^n$ transition matrix of a Markov chain
Sorry, will try to add details to clarify.