Mathematics

Q: Moment generating function of multivariate normal distribution using mgt of standard normal distribution

Hi. Could you please look at the question I just posted? Thank you

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The problem I am looking at asks to evaluate the moment generating function of the multivariate normal distribution $N\sim(\mu,\Sigma)$ using moment generating function of the standard normal distribution. I have the followings in my notes: For $k$-dimensional $\mathbf{X}$ on $(Ω, F , P)$ with d...

12:33 AM

I'm not a probability expert, but can't you just apply $e^{u+v}=e^u e^v$ and separate the integrals?

@Agi Ted is correct

Score 1 for Ted.

@TedShifrin I know you're rusty on measure theory, but just thought I'd give this a shot: define the Borel sets in $\mathbb{R}$, $\mathcal{B}(\mathbb{R})$, as the $\sigma$-algebra generated by all open sets in $\mathbb{R}$. How would you show that it has cardinality equal to the continuum? I don't even know where to begin.

(not homework, just studying)

It's not just that I'm rusty. I don't like the stuff. But there are several people in here who can do this in a second (e.g., @Thorgott @Alessandro).

I'm just looking forward to finally doing some probability stuff... it has been a headache to learn about regularity of measure over the last few weeks

12:38 AM

@Clarinetist Do you know how to build it "layer by layer"?

@AlessandroCodenotti No idea what you're talking about.

Countable union of countable unions, etc.

uhh, transfinite induction?

So when you say "the $\sigma$-algebra generated by the open sets" you have the whole thing at once

@TedShifrin Even if I do that, I'm not sure about the $dF(x)$ part, since I think the variables are not considered independent from one another. I also don't get where should I use the mgt of std normal

12:39 AM

I looked up what transfinite induction is, but I wouldn't know where to apply it

@Agi They must be independent for that factorization to work with the MGFs

Why are the variables $(x_1,\dots,x_k)$ for $\Bbb R^k$ not independent?

But you can start with open sets as the first layer, take union of complements as the second layer, unions of complements of things in the second layer as the third one and so on

OK, Clarinet can help Agi and Alessandro can help Clarinet, and Ted — thankfully — can be lazy.

@Clarinetist so I can assume they are independent?

Then the proof boils down to showing that layer one has cardinality $\mathfrak c$, moving up from one layer to the next doesn't increase cardinality and the slightly harder part is that there are at most $\omega_1$ layers

12:40 AM

The only explicit description of the Borel-algebra I know is by transfinite recursion, so I'd try proving the cardinality doesn't exceed that of the continuum by transfinite induction - I haven't actually done this before, tho

(the proper name is the Borel hierarchy for what I'm talking about)

@Agi Yes, I believe it can be shown that product factorization on the MGF of the sum holds if and only if those variables are independent

(considerably harder is showing that, in this case, there are exactly $\omega_1$ distinct levels, but luckily that is not needed)

@AlessandroCodenotti Never heard of it until now, thanks. My professor had given it to us as an exercise, and I suspect he may throw it on our exam. As far as I can tell, no one in my chat group for my class knows where to begin.

@Clarinetist Oh yes for product factorization they should be independent but I am not sure if they are here since there is no statement saying they are independent.

12:43 AM

@Agi Actually, it does - it says in the problem that $X$ has independent components

@Clarinetist That was just what I found in my notes, the question is just to find the mgf of multivariate normal dist using standard normal mgf

that sounds like a rather hard exercise, unless you're doing something more set-theory oriented

@Thorgott Definitely not. That was brought up as an exercise in our first measure theory lecture.

@Agi The very definition of the multivariate normal distribution requires that each component of the vector can be written as a linear transformation of independent $N(0, 1)$ random variables

@Agi Let me give this some thought quick

northerner

I was wondering if someone can provide intuition as to the difference between $f(x+1)$ and $f(x)+1$? If you have a function $f(x)$ and it's inverse $f^{-1}(x)$ is it possible the inverse of $f(x+1)$ is $f^{-1}(x)+1$?

have you tried some examples

12:50 AM

@Thorgott I'm not sure where one would even begin by way of "examples"

that was to northerner

as for your exercise, I can not think of a way of tackling it that isn't built on using ordinals/transfinite induction or something along those lines

which is why that exercise strikes me as rather odd in a general measure theory course

@Clarinetist This gives some insights but I'm still confused on how to begin. courses.washington.edu/b533/lect4.pdf

@Agi Sorry, I realize now that question isn't as trivial as I thought it'd be. So first things first, you're probably going to be dealing with using the fact that multivariate normal distributions can be written as a linear transformation of standard-normal random vectors

@Clarinetist yes

@Clarinetist In the document I sent, would proof def3 to def2, in the second page, suffice?

@Agi Oh. I think this would be the method of proof: let $\mathbf{Z} \sim \mathcal{N}(\mathbf{0}, \mathbf{I})$. Find its MGF. Then apply linear transformations on MGFs. Then you'll reach the result in that document.

@Agi Yeah, it is more or less that.

I haven't had to think about multivariate normal distributions from first principles in a while, thanks for asking that.

12:56 AM

@Clarinetist Sure thing, thank you too!

@Thorgott I think I'm just going to bet on the professor not asking that on the exam at this point. I agree it's silly to put as an exercise.

The solution I found deals with uncountable ordinals... I don't even know what that means

Good book. Math art dianadavis.github.io/im/book.pdf

"Book". PDF

@Agi One of the things I've learned when it comes to multivariate normal distributions is that 95% of the time, direct integration isn't the way to do work with them.

Well, you actually can get a physical copy here if you want bookstore.ams.org/mbk-135

but it's mostly pictures

@Clarinetist indeed

1:00 AM

Also

Hyperbolic Paper Demo

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1:55 AM

This is a question about transformation matrix. After watching 3blue1brown videos, I've assumed that a 2x2 transformation matrix has a 2 vectors—i hat and j hat. i hat lies on the x axis and j hat lies on the y axis. what about 3x3 matrix?

Trefoil knot in KnotPortal

Order 2 branched cover of the trefoil knot:

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do i hat lies on the x axis, j lies on the y axis and k hat lies on the z axis? Also what is the convention for this axises? Example: the x axis is pointing to my right, y as my up and z as my front. Is that correct or z and y are swap?

Lucas Henrique

2:17 AM

hey chat

when it comes to matrix Lie groups, can you generally speak about the connectedness of $\mathrm{GL}_n(\mathbb{C})$?

> There seem to be two kinds of geometers: those who imagine that their
mathematical objects are something that could fi t on your desk, and those
who imagine their mathematical objects are room-sized, something you
could travel through.

Lucas Henrique

I read a few questions about topological aspects of this group, but I'm not sure if they hold independently of the topology

- Sebastian Bozlee (though the idea might be older)

It's connected in the usual topology induced by $\mathbb C^{n\times n}$, and I believe it's connected in the Zariski topology, but you can certainly give it the discrete topology or something and make it disconnected

Right?

Lucas Henrique

I don't know, because a Lie group requires that the group is a differentiable manifold

Oh

"Right hand rule" @Spade000

(I like to use movement - have z be my thumb, but curl the rest of my fingers together so they all start at x and end at y)

@LucasHenrique I think the topology is induced by the Lie algebra

Hm, maybe not

2:24 AM

I feel like this might be a subtle question

No wait, it's easy

No wait, it's not

lol

I mean, different topologies give different tangent spaces and thus different Lie algebras, right?

If you have the Lie algebra you have the Lie group

But maybe you can have two Lie groups that are isomorphic as groups but not (topologically) homeomorphic

you mean different smooth structures?

it's a fact that if a topological group has a Lie group structure, that Lie group structure is unique up to diffeomorphism

but you might be able to equip a group with two different topological group structures, which in turn admit different Lie group structures

The question is about different topologies

Yeah

discrete topology would give an easy counter-example if we don't require manifolds to be second-countable, but alas, we do

this is awkward

@AkivaWeinberger So z does seem pointing up instead of y. What about these 3 vectors i hat k hat and k hat? I mean not lie groups. In this rotation matricea wikimedia.org/api/rest_v1/media/math/render/svg/… do i hat represent the x axis, j hat as y and k hat as z?

2:29 AM

i is x, j is y, k is z

You can rotate your hand so that y is up

"x towards me, y up, z right" follows the right hand rule

So does "x right, y up, z away from me"

"x east, y up, z north"

Wait did I do that right

No I didn't

Swap x and z all of those

@AkivaWeinberger that confuses me as y is pointing left.

in the picture

If you rotate your hand it still follows the right hand rule

There are only two possible orientations for a coordinate system: right handed and left handed

You can't rotate your right hand to look like your left hand, so these really are different

All three of these are right handed (since they're rotated by 120 degree rotations about the line x=y=z)

There are 6 logical possibilities for permuting three objects

3 are right handed, 3 are left handed

the question reduces to: given a group, are there two topologies both turning the group into a topological group and into a topological manifold, yet such that the resulting structures are not homeomorphic

Q: Does a Lie group's group structure (not Lie group structure) determine its topology?

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Question in title. Said another way, can you have two Lie groups that are isomorphic as groups but not homeomorphic? If so, the group isomorphism map will not be continuous (and thus not a Lie group isomorphism), and there will be no natural map between their tangent spaces (Lie algebras). I susp...

general-topology group-theory lie-groups

Asked main

It seems to me that there are no convention, as far as I know, where x, y and z is pointing in the real world. But I'll stick to the right hand rule. @AkivaWeinberger thanks for you accompany.

2:43 AM

nice

I want a 2D video game that takes place in the universal cover of a disk minus the origin (perhaps stylized as a top-down view of a spiral staircase). The branch cut can be located opposite the player

Spiral staircases solve VR's locomotion problem, actually

the problem of walking in the virtual world while not crashing into walls in the real world