00:00 - 12:0012:00 - 23:00

12:00 AM
@MikeMiller I can delete a link after 30 seconds so are you quick? :P

Yeah he and Venjakob do a bunch of p-adic Hodge theory and Iwasawa theory and the $(\varphi, \Gamma)$-modules that I wanna learn about
lol

that's a very scary sentence

@Thorgott this

@user2103480 Go

here ya are

12:02 AM
One of the doctoral students said in an email: in many senses p-adic Hodge theory is just complicated linear algebra

@EdwardEvans thats what almost anybody says about almost any subject

hahaha

I don't know MK but CW is a nice guy

What if linear algebra is already complicated linear algebra to me

there you also see his topology I/II courses

12:04 AM
@Alessandro this was my reaction

@MikeMiller I think you can "him" him :P he was an assistant at my old uni so some of my friends are personally acquainted with him

legend has it that mathematicians to this day are still trying to come up with a new subject, but just keep accidentally reinventing linear algebra

One day we will understand nonlinear algebra, but it will be a weapon too terrible to use

@Thorgott linear algebra LXXXVII: symplectic boogaloo

I like how the time of most activity in chat is 2am in European time, but just because we don't sleep rather than because that being when all the US people are online

12:08 AM
F

sLeEp

if we give mochizuki the naming rights it'll be intergalactic boogaloo

lmao

I gotta be awake by 11am
s'all good

Oh Thomas got a job there?
That's great! I love Thomas
(Saw on CW's webpage; I mean TW)

12:10 AM
Ahh

@Alessandro how was your AlgGeo course? I remember you saying you didn't really like it, why was that?

HU seems to be quite good in geometry. At least all the lectures have weird names

Hahaha
It's at least filled w people in my field
So I can't complain

@Thorgott also, reinventing infinitesimal stuff is a common thing

difference between continuously varying group and Lie group?

12:15 AM
@EdwardEvans Turns out schemes are not my cup of tea

infinitesimal stuff? you mean non-standard linear algebra?
@geocalc33 one of them was made up by you, the other was not

@Thorgott got me there, shoot

no no non nonnonon

@Alessandro I'm somewhat afraid of that lol

but I actually meant linear nonlinear algebra

12:16 AM
I didn't make up "continuously varying group" I swear

AlgGeo will follow basically directly from Algebra 2 and the Algebra 2 course was hard af

What was algebra 2?

@geocalc33 tell that the prosecutor
@AlessandroCodenotti a continuously varying group of courses under that name

Well, there was a section on commutative algebra which was fine, but the other two sections were tons of category theory and homological algebra lol

@user2103480 sorry I only know about (strongly) continuous semigroups

12:19 AM
what's not to love about that

So my issue with schemes was that I don't have the background to appreciate them

@AlessandroCodenotti I love that your math education is as erratic as mine

I think I'm just gonna find it hard rofl
I've been skimming over Ravi Vakil's notes and most of the motivating examples in those notes come from differential geometry and I don't know any differential geometry rofl
even though the examples make sense "somehow"

Like I could follow the technicalities, but after a semester it felt like I knew a lot of definitions and weird properties a scheme can have, but I wasn't really able to solve any new problem that I couldn't solve before knowing about schemes so what was the point?
Like imagine doing a first course in general topology in which you go through all the various weird and ugly properties spaces can have, without ever seeing a single example of why topological spaces are a powerful idea, something like that

@user2103480 Everyone's is
If they say otherwise they're lying
Perhaps to themselves

12:25 AM
I've seen enough people settle pretty quickly on one area

I would tell the prosecutor that I'm talkin' bundles. continuously varying group in the sense that the groups vary continuously over a line. Each fibre is isomorphic

so you want a principal bundle or?

@AlessandroCodenotti all too often the actual core questions and goals aren't presented well

@Thorgott yeah

the goal of algebraic geometry is to do cool algebra and lure people in by pretending it's geometric
or so I've been told

12:29 AM
lol

The way PDEs and differential equations are often introduced is totally devoid of physical intuition

I think I'll just read Hartshorne cover to cover and then pretend I watched the lectures

No don't, chapter 1 of Hartshorne is atrocious and presents stuff which is usually not even done anymore in courses. Start at schemes

@Thorgott linear algebra actually

@Alessandro I'll learn about varieties anyway from the seminar on elliptic curves

12:31 AM
What does the divergence of a vector field mean here? What's this and that? Too often the intuition actually lies in a somewhat intuitive integral equation, which is equivalent to the pde under this and that condition

Where else would one learn about varieties

Linear Algebra: An Algebraic Approach

The Royal Variety Performance is a televised variety show held annually in the United Kingdom to raise money for the Royal Variety Charity (of which Queen Elizabeth II is life-patron). It is attended by senior members of the British Royal Family. The evening's performance is presented as a live variety show, usually from a theatre in London and consists of family entertainment that includes comedy, music, dance, magic and other speciality acts. The Royal Variety Performance traditionally begins with the entrance of the members of the British Royal Family followed by singing of the national anthem...

I suspect one could read ch. 1 only when he backreferences it

so you're saying one shouldn't read hartshorne linearly?

12:34 AM
I would say it shouldn't be read at all but that's just me

Vakil's notes seem entertaining and my tutor for algebra 2 won't shut up about them

@Edward I'll come crying to you when I do AG next semester

okay I am understanding it more now. You can have a principle bundle and a Lie group $G$

@Thorgott I'll cry at you because I failed AG

@AlessandroCodenotti people say that about some of the all-out combinatorial war in some set theory II lectures

12:36 AM
we can just cry in unison then

Vakil's note saved me when I took AG

The Rising Sea is a reference to a sea of tears

Is it a rite of passage to get one's soul crushed by AG?

idk but I'm preparing for it like a doomsday prepper

because I don't intend to go there

12:38 AM
being a rite would imply that it ends

This is your last chance. After this, there is no turning back. You take the blue pill,
the story ends, you wake up in your bed and believe whatever you want to believe. You take the red pill, you stay in Wonderland and I show you how deep the rabbit-hole goes.
this is the first quote in his notes
rofl

based
insert deep fried ravi vakil here

lmao

if a Lie group is a smooth manifold, then by varying it using principal bundle techniques, you'd be varying the smooth manifold. So what? You'd have some isomorphic manifolds
why is this useful

12:43 AM
yes

@Edward just read EGA, avoid the adaptation and go for the original

a fine idea

@MikeMiller so, should I care to learn much simplicial homology beforehand, or should I directly dive into singular homology if I want to prepare - as taught in the book by fomenko & fuchs for example?

You should read the first 30 pages of Dieudonne, "History of algebraic and differential topology", and then read whatever you want for the formal junk
Everyone ought to know how Poincare thought of it

Fair enough, that sounds like a deal!

12:54 AM
When you learn singular homology you should try to understand how the formal junk relates to Poincare's idea. For this you should either ask me for pointers or read the proof that pi_1^ab = H_1 at the end of Hatcher 2.A, where the idea is reasonably clear
Nobody writes about this from the beginning since it's too much work to introduce a historical but incorrect idea first
But it's where most of the intuition comes from

thanks! that's a large book haha
even in terms of MB
knowing the guiding ideas is pretty important to me so I'm sure it's a helpful recommendation

MB = ?

sorry, I shortly thought about clarifying that I mean megabyte, note some person :D

Dieudonne's book is a great resource, I learned a lot from it
I also really like Kreck's book "Differential algebraic topology" if you know some smooth manifolds / transversality. But it maybe isn't the right place to learn this for the first time

I should try reading some of that, Kreck is fantastic

1:06 AM
I need guidance of root of unity
confused how the book did it.
1

Let m and n be positive integers have that have no common factor. Prove that the set of numbers $(z^\frac{1}{n})^m$ is the same as the set of numbers $(z^m)^\frac{1}{n}$.We denote this common set of numbers by $z^\frac{m}{n}$. Show that $$z^\frac{m}{n} = \sqrt[n]{|z|^m}\left(\cos\left(\frac{m}{n}... Consider two sets A, B ⊆ R such that A is not a subset of B and B is not a subset of A. If sup(A \ B) ∈ B and sup(B \ A) ∈ A, prove that sup A = sup B. Proof. (A \ B) and (B\A) are non empty, we know that A\setminus B \subset A so,sup(A\B) \leq sup(A) we know that B\setminus A \subset B so,sup(B\A) \leq sup(B) This much thing Icould drew from the question Can you give hints? 1:33 AM @MikeMiller CW complexes make more sense after reading the passage about the types of spaces poincare intended to consider 2:16 AM Hi, I'm confused about a thing in mutlivariable calculus. I want to use the Hessian test for characterising a critical point of a function say f(x,y) \to \R. Assuming that the function f is much easier to understand via polar coordinates r,\theta can I just compute the matrix of second derivatives w.r.t. r and theta and use the usual hessian test applied to that matrix? what confuses me is that I know I should use the chain rule in order to pass from the usual Hessian in cartesian coordinates to the polar coordinates representation but I cannot think of a counterexample of a function f such that in polar coordinates the matrix of mixed second derivatives would give me a different information from the real Hessian in cartesian coordinates 3:13 AM @Luigi What about the simplest example? f(x,y) = x^2+y^2? @EM4 The point is that because m and n are relatively prime, the integer multiples of m give all the possible remainders mod n, so km is just as good as k as k varies over integers. Thanks for sharing this Professor @TedShifrin Hi @robjohn :-) 3:32 AM @skullpatrol hey there How's it going? @robjohn @skullpatrol pretty good. Running from one fire into the next. This is fine. ^that will automatically pop-up the meme, in some chatrooms @TedShifrin in this case f(r, theta)=r^2, but I'm not really sure what to conclude here because you cannot really derive at r=0 am I wrong? @TedShifrin in any case, if I'm following your reasoning the matrix of partial derivatives would be semidefinite positive if we use polar coordinates, and not definite positive in the case of cartesian coordinates. is that the point of the counterexamples? 3:56 AM Yes, Luigi, so your version of the test fails to show a local min. I see. thanks! @skull: Did you study it? :) I tried Tried? Did I mess up? At first I thought you did. 3:59 AM Hmm. But it works. LOL, so I'm confuzled. Is there something I should try to explain? It would be a good question for AoPS. How so? What's the question? Not really sure... 4:04 AM I've done that both with high school kids and with middle schoolers. It was too tough for the middle schoolers, but I had them trying to decide one of my favorite problems. Hmm, perhaps the lack of trig/geometry in middle school had something to do with it? Yeah, the kids were a bit younger than I had been told. But we had fun anyhow. So, are you done with the AoPS? or even their Beast Academy Please consider joining their FB group :-) 4:33 AM @skullpatrol link? 4:46 AM @skull: Yeah, I'm done, although I have written several letters of recommendation for previous students. It was never the best fit for my style of math or teaching, but I'm glad I did the two years. ok would zoom webinars be more suitable? I'm sure there's enough interest in this room to set up something :-) Just a suggestion, of course. Interest in what? 4:58 AM Any topic you choose, Professor Me, nah. \Phi(s)=\Gamma\left(1+\frac1s\right)+\sum_{n=0}^\infty\frac{(-1)^n}{n!}\zeta(-ns). This can't have any zeros because of the gamma function part right? For a vector bundle \pi : E \to M my problem set talks about \pi^{*}E. What do they mean by this? At first I thought pullback bundle but don't you need a function using which you take pullbacks? 5:18 AM Hey everyone! Notably @Ted! Haha. Hi, Demonark. @Sayan You have a map. They wrote it! I answered a question of one of your UC classmates, Demonark, who is now at Stanford. @user2103480 Aha right Heya, a Balarka! Hi Ted Hmm so it's the same as the vertical bundle right? Hey @BalarkaSen 5:32 AM yo, Balarka The fibers are tautologically the fibers of E. @Sayan Vertical subbundle of TE if you want to think about it that way, yes. I like Ted's tautological comment. Ted is good only at tautology. Good mathematics should be mostly tautologies I hope not. 5:36 AM @TedShifrin Ah nice That's what gives it the reputation of being "memorizable." Where was the question? I'm mildly curious lol Let me hunt a sec. Here. I assume you know this person quite well. Ah yeah, he was one of my bootcamp mentors He never let on that he knew my name, though :P 5:42 AM To be fair I think he was an analysis mentor rather than a geometry mentor So he focused more on the complex analysis and probability parts of the bootcamp. Though he did the bootcamp himself the year before I presume and they still used your book Well, he seems to be doing geometry now. Ah, I see your point. Bootcamp: A geometric approach :P Yeah looking at his page he's into different things now than I remember In undergrad he was more of a Schlag harmonic analysis guy I should ask my good friend at Stanford (who's dept head during COVID) if he knows him. Haha, maybe! I will say it's mildly comforting that he doesn't yet have grad papers lol 5:49 AM Isn’t he your year? Yeah Idk I've been worrying recently that I'm kinda behind in a way, like if a good block of this year is spent getting background then I've barely got 2 years to do research before job apps These days most people don’t finish in 4. Well I was thinking 5, just that job apps are done during your 5th year, no? Princeton used to kick people out after 3. Yes, fall of 5th. So I'm assuming that the bulk of the research will happen 3rd and 4th year, and that early 5th year you're mostly trying to get stuff submitted and apply to jobs. So yeah I'm just hoping I can do something during that time lmao 5:54 AM I have a probably dumb question. Let A be a finite-dimensional semisimple algebra over an algebraically closed field of char 0. Is the dimension of the center of A equal to the number of simple modules? If so, how to I see that? Interesting how so many MSE stars left academia. Balarka mentioned this. Qiachu, Zev is another (from Chicago). I wonder what's become of anon. Yeah I'm not sure about anon lol I wonder if it's burnout Qiachu, Zev et al are active on Twitter 6:25 AM ice wizards aren't as good as the normal wizards pro farming technique is to max army camps with baby dragz and surround the TH. guaranteed 1 star plus gold and elixir 7:03 AM @TedShifrin one thing I'm wondering, since you emphasized participation in your classes How did you handle it when people were just super quiet? hi, folks, I have a problem as follows. 0 Background It will be easier if we distinguish "mutually exclusiveness" from "independency" by considering the sample space in mind. Two events that are compared for mutually exclusiveness must be from a single sample space. For example, Tossing a coin twice. A=\{HH\} is an event in which t... 2 hours later… 9:22 AM Regarding prime numbers I found the following formula: \Delta\varepsilon(n)=\frac{1}{2}{p}_{n-2}+\sqrt{-\frac{3}{4}{p}_{n-2}^{\:2}+{p}_{n-1}^{\:2}}-2{p}_{n-1}+{p}_{n-2} \Delta\varepsilon(n)=1.5p-2(p+g)+\sqrt{-0.75p+(p+g)^{2}} Where \Delta\varepsilon goes to 0. According Wolfram Alpha this expression can be written as the following series: \Delta\varepsilon(n)=-\frac{3g^{2}}{p}+\frac{12g^{3}}{p^{2}}-\frac{57g^{4}}{p^{3}}+\frac{300g^{5}}{p^{4}}-\frac{1686g^{6}}{p^{5}}+\mathcal{O}\left( \frac{1}{p^{6}}\right) 10:14 AM @TedShifrin did these people share reasons? Apart from qiaochu, whose reasons I could google @Balarka @Alessandro just got my talk: I'll be doing tons of analysis on locally profinite groups! which might have been a lapse in judgement What's a locally profinite group? Hausdorff, locally compact, totally disconnected topological group It's just a profinite group but instead of compact it's locally compact I guess I've seen the "realization" that "it's of all of no use" in some of my peers, especially some of the focused and successful omes My talk will be "The L-Function and \varepsilon-factor of a cuspidal representation of \operatorname{GL}_2(F)" 10:26 AM @EdwardEvans Oh ok, seems interesting yeah the amount of analysis might kill me but it'll be good practice rofl for when I actually die @BalarkaSen Grothendieck has entered the chat @user2103480 CW complexes are introduced in a way that's too complicated A CW complex is built up out of discs. Triangulable spaces are CW complexes but so are many decompositions into "cells" which don't glue together quite as nicely EG, CP^n is a CW complex: the open parts of its cells correspond to C^k = {[z_1 | ... | z_k | 1 | ... | 0]} Did we actually all just sleep about 8 hours and are now back here @user2103480 literally I just look at this tab now and again 10:30 AM Fair enough I guess the second lockdown can come if that is my life now Are you a citizen of the US? lol^^^^^^ Anyway, @MikeMiller, in the book, do you mean that I should start when dieudonne explains poincarés analysis situs, or did you also mean to include the introduction? @EdwardEvans me? Nah, germany Ahhh okay, same @user2103480 8 hours? Look at this optimistic guy 10:33 AM Because the introduction goes over my head pretty quiclly @AlessandroCodenotti it was nearly 4 to nearly 12 for me @EdwardEvans Are you a German citizen? Oh, not yet, but I'll be abandoning my British citizenship as soon as I'm allowed can I have it yeah it's in the bin behind McDonald's can you sell it to me 10:35 AM it's not worth anything but sure @EdwardEvans Time to get married with a German then I guess rofl, my girlfriend will be pleased to know that I'm only with her to get a German passport I think he just needs to live here for long enough :D Hi chat @user2103480 I think the usual time is 10 years but reduced to 6 if you can show significant integration or smth for some appropriate definition of significant integration 10:39 AM Astyx hi @EdwardEvans Good thing you're doing analysis then weeeyyy @AlessandroCodenotti you mean "job-mathematics"? it's analytic number theory though, which is enough to tempt me Honestly, many companies dont even care. But these are mostly insurance companies and such, or software developement jobs without a big mathematical component Which is reasonable enough, at least theres always a well-paying boring job around the corner lol 10:43 AM if mathematics doesn't work out I might just try and get a job in some crypto position and say I did a seminar on elliptic curves is that wallpaper I wanted to say that you seem to ditch insurances in favour of intelligence services but that might actually be hard without citizenship yeah is there a reason why the elliptic curve is wrapped around the torus? (Ofc theres enough other crypto work) 10:45 AM Elliptic curves are complex tori @user2103480 yeah, Darmstadt is pretty nearby and apparently Darmstadt is big on crypto so I'll just go there and pretend I know what I'm talking about @user2103480 I forget what's in the introduction. I just meant analysis situs, up until he diagonalizes some matrices and obtains "torsion numbers" and a little bit after that Noether turns these into groups Anyway I forgot to finish my thing about CW complexes They're literally made to run inductive arguments CW complexes have a topology so that a map is continuous on the CW complex iff it's continuous on each closed cell So if you can compatibly define a map on all the cells you have a map from your CW complex One usually starts from points and then shows you can extend up to the 1D cells etc @EdwardEvans darmstadt also has a good logic group if you're interested in that I absolutely am not but thanks understandable, have a nice day hahaha danke I don't do mathematics for the logic, I do it for the symbols that make people think I'm clever 10:53 AM Halfway through the Rivers of Nihil album I got an ad for ion chromatography equipment on youtube, I'm quite confused @Edward dafuq nice album though, right? yep, I like it so far @EdwardEvans Don't we all universal truth for the algebra people ^ 11:07 AM @MikeMiller found it btw Hi there. I got a pretty basic question about stochastics but somehow I can't seem to figure it out: Let X_1,\dots,X_n be i.i.d. according to a distribution with density function f(x). Let M(x) = \max_{i=1,\dots,n} X_i. How can I calculate the expected value of M(x) with respect to f(x)? My problem is the "with respect to f(x)", so the expected value is \int M(x) f(x)dx. If it wasn't specifically with respect to that I would just need to calculate the density function h(x) of M(x) and the expected value would be \int M(x) h(x) dx. Can I still do it like that? The expected value isn't well-defined if f is a density R^d -> R, since M doesn't depend on X, it depends on an underlying probability space Is it an exercise? If yes, whats the exact statement? *M doesn't depend on (small) x 11:25 AM 0 Consider a commutative non-associative ring A of finite dimension, that is power-associative. Now define weakly algebraically closed in T as :$$ x^2 + a_1 x + a_2 = 0  always has at least one solution $x$ that belongs to $T$. ( the coefficients $a_1,a_2$ are in $T$ ) So I wonder : When is...

@LeakyNun I don't get what's wrong with the proof

@user2103480 Could you please evaluate what exactly you mean by saying that $M$ doesn't depend on small $x$ but on an underlying probability space?
@user2103480 Nah it's not an exercise, I need it for my thesis. More exactly what I am trying to check is whether $\mathbb{E}_{\beta}[M(x)] < \infty$ where $\mathbb{E}_{\beta}[M(x)] = \int_{- \infty}^{\infty} M(x) p_{\beta}(x) dx = \int_{- \infty}^{\infty} M(x) p_{\beta}(x) dx$ with $M(x) = \max_{k=1,\dots,p} \frac{X_k^2 e^{X^T \beta}}{(1+e^{X^T \beta})}$

@Astyx How to justify "each elt of C is a lin comb of the y_i with coeffts in B_0"?

and $p_{\beta} = f(x) \cdot \frac{1}{1 + e^{x^T \beta}}$ where $f(x)$ is the distribution of the standard normal distribution according to which $X_1,\dots,X_p$ are i.i.d.

@MathStudent "small x" just meant that I don't mean a random variable but a real number/vector
M is defined as the maximum of n random variables which (hopefully) are defined as functions from a probability space to R^d, d>= 1
so M is a random variable as well
beta is a constant?

11:31 AM
So you have $c\in C$ written as $c = \sum_I b_I y^I$ where the sum is over tuples of integers lower than a certain $N$, of length lower than a certain $M$
Does induction on $M$ not work ?
Is the issue with constants ?

@user2103480 Yeah, $\beta \in \mathbb{R}^p$ is an unknown vector of parameters but I think for this, it can just be seen as a constant

You're coming from statistics, huh? :D

@LeakyNun What am I missing ?

@Astyx yes
they forgot to check the base case

@user2103480 At least in the US we often teach calculus-based probability before anything measure theoretic. Students think in terms of the PDF and evaluate probabilities by integrating over subsets of R^n (aka pushing forward the measure)

11:36 AM
"Each element of C is an affine combination of the y_i with coefficients in B_0" ?
Hence C is finitely generated as a B_0-module

@user2103480 If you mean statistics major, then no. Embarrassingly, I am a math student hopefully about to finish my bachelor's with some kind of severe "knowledge holes" in basics. But yeah my thesis topic is from statistics

Oh wait no
Ok I see it
Does stating that A, B, B_0 and C share the same unit not work ?

it doesn't

@MikeMiller Ah well that makes it harder to explain where the substitution went wrong

Why not ?
$\{1, y_1, \dots, y_n\}\subset B_0$

11:39 AM
@MikeMiller Not sure what you mean exactly but you're right about me having learned probability theory after having learned measure theory
Because we were able to use measure theory for the proofs and it was a still a very fast-paced lecture

@Astyx then you're modifying the yi's

Yes

sure, then that works

My argument is that having an affine combination of the y's is sufficient to arrive to the same conclusion
Because the unit is the same at every level

I don't understand what you mean

11:42 AM
@MathStudent essentially, you want to subtitute all n random variables so the "right" integral you would obtain here would be an integral over R^n of M(x_1,...,x_n)*f(x_1)*...*f(x_n) dx_1, ..., dx_n
But that's not completely right

what does affine combination mean?

since I don't know how the densities behave depending on the parameter beta

$c = b + \sum b_i y_i$

I'm just assuming that for all beta, there is a density

I shouldn't have deleted my messages
I still stand by them
The final claim is not that $C$ is generated by the y's, so it's ok to throw 1 in there too

11:45 AM
@user2103480 Interesting. So you are saying that I can't do it the same way I could if I wasn't specifically looking at the expected value w.r.t. $p_{\beta}$ but differently then, right?

@MathStudent Nah I think the measure theoretic stuff makes more sense. I'm glad you have the background in it

You said you're German, right? This website peter-junglas.de/fh/vorlesungen/stochastik/html/kap1-5-2.html shows how to calculate the density function of the maximum of i.i.d. random variables and with it the expected value. That's how I would think to do it if I didn't have to calculate the expected value w.r.t. $p_{\beta}$ and hoped i can still do

@Astyx yeah I never heard of the term affine combination so I thought you just meant linear combination

@MikeMiller in his lecture on the twelve problems of probability, rota actually argued for reviving that tradition in "higher" math as well

@user2103480 Yeah, you can assume that

11:46 AM
Not sure it exists, but I mean it as in c is an affine function of the y's
Ah damn, it actually means something else, sorry

I really don't like the calculus based language
I was always very confused about it
Easy to confuse an RV and a PDF

1

Convolutions can be expressed as a matrix-multiplication (see e.g. this post) and as an element-wise multiplication using the Fourier domain (https://en.wikipedia.org/wiki/Convolution_theorem). Attention utilizes matrix multiplications, and is as such $O(n^2)$. So, my question is, is it possible ...

It could be interesting to some of you.

@MathStudent I can't say that you can't since I really don't have all the information, but in the setting of i.i.d. variables X_n, and the maximum function M(x_1,...,x_n), the expected value E[M(X_1,...,X_n)] can only reasonably transformed to a real integral by simultaneously substituting the vector (X_1,...,X_n)
Then, since all these are i.i.d. and admit a density, the joint distribution of (X_1,...,X_n) has density f(x_1)*...*f(x_n), where f is the density for those RV
@MikeMiller ditto. We got our asses kicked by our probability I instructor that basically made three quarters of the lecture about measure theory (which I forgot all about), but it was well worth it

@MikeMiller Ah, I see that you meant it the other way around now thinking I didn't learn measure theory first and thinking of probabilities in terms of integrals because of that. Somehow, I sometimes prefer that kind of language anyway. I guess it seems more precise to me sometimes and I wouldn't really know how to check whether the expected value is finite otherwise in this case. I guess maybe it has to do with my measure theory knowledge not being good enough

Don't take my comment as an indictment! It's my personal taste. I just never understood well in the calc context.

11:55 AM
@MikeMiller And in a previous stochastics course, it was half measure theory, half "this somehow works", and I was often lost at a point where I knew what I should do intuitively, but lacked the mathematical tools

Gotcha, I didn't take it as such, no worries.
@user2103480 Do you know what I would need to substitute it by? And how would that help to transform it to a real integral?

Is this valid? $\int_{-\infty}^{\infty} X_k^2 e^{{{X}^T}\beta} f(x)~dx=E_{\beta}[M(x)]$

@user2103480 How would the joint distribution come into play with the expected value of M(x_1,...,x_n)?

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