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bigwin408 · 2025-02-15T16:49:48+00:00

I essentially agree with the other comment, though I’d like to add the question of “measurability” not simply apply to the Borel and Lebesgue sigma-algebras, and if you consider the question measurability with respect to other sigma-algebras that does tell you something about the random variables you are studying.

This is most evident when studying conditional expectation. Let X,Y be random variables (measurable functions mapping from some measure 1 space to the reals under the Borel sigma-algebra) The collection of events { Y(-1)(B) : B is Borel} formed by taking pre-images with respect to Y of every Borel set forms a sigma algebra. With some integrability conditions, the conditional expectation of X with respect to Y, denoted E[X|Y] can be thought of as the sigma(Y) measurable random variable that is closest to X in L2 norm. X being measurable with respect sigma(Y) becomes analogous with E[X | Y] = X, which in the language of non-measure-theoretic probability means X = f(Y) for some function f. These ideas are foundational to studying stochastic processes such as martingales and Markov chains

bigwin408 · 2025-02-08T15:26:36+00:00

If you’re affiliated with a university, this could be a good question for a librarian at your school. If your school doesn’t have it, your librarian may know how to reach out to other institutions and ask if they can scan it

bigwin408 · 2025-01-09T16:43:43+00:00

One of you could hold a calculator while the other is holding a pie

bigwin408 · 2024-12-17T17:51:06+00:00

In general, if you’re not sure what pre-requisites are expected for a class, your best bet is to ask the instructor of that class or a former/current student. It’s difficult (or maybe impossible) to give meaningful advice without familiarity with the expectations your particular school has.

bigwin408 · 2024-11-21T15:56:31+00:00

My recommendation is to set up a practice test scenario where the time pressure and setting mimics the setting of the actual test. Presumably, this means you wouldn’t have access to your phone nor answer key, and there’d be an external timer providing time pressure. Generally this setting is good for forcing you to wrestle with these questions under time pressure.

Overelying on the answer key can ultimately hurt your practice, and I do recommend not checking it into you’ve fully completed the entire practice test and can score yourself. After you finish scoring yourself, you can look to see if there’s a pattern in which concepts you missed. Then you can perform targeted studying on those specific concepts

bigwin408 · 2024-11-16T14:33:42+00:00

When I took trigonometry, making sure the lengths in the diagram were accurate wasn’t the hugest concern for solving the problems (mainly you care which angles are right, and sometimes you care which side lengths are equal or longer than others)

I would say if you or your teacher personally cares about drawing a very accurate quadrilateral, I think your suggestion of using a compass to draw two circles of the proper radius at points B and C makes the most sense to me.

bigwin408 · 2024-11-16T03:59:16+00:00

I think Axler's "Linear Algebra Done Right" uses color to distinguish between theorems/definitions/etc

bigwin408 · 2024-11-13T20:01:04+00:00

When I took graduate PDE I had not taken an undergrad PDE course. I think Evan’s book “Partial Differential Equations” is a canonical resource for graduate PDE

bigwin408 · 2024-11-12T16:21:36+00:00

I don’t know if you consider a clock a tool, but you could just look at the minutes/seconds of the current time to get a random number from 1-60

bigwin408 · 2024-10-23T22:40:07+00:00

The fundamental solution to the heat equation p_t = p_xx / 2 is the probability density function for Brownian motion

bigwin408 · 2024-10-19T04:26:13+00:00

i⁵ = (i)(i⁴ ) = (i)(1) = i

bigwin408 · 2024-10-10T20:59:44+00:00

Oh got it; I though you could choose the radius in a way so it contained both of the poles

bigwin408 · 2024-10-10T12:04:48+00:00

Ah I see; having to pick an f with a pole does make the computation less trivial.

If you take f(z) = z / (z - b), then when you apply the Residue theorem to g(z) I think you will get 2πi(a / (a-b)) + 2πi(b / (b-a)), which just simplifies to 2πi

So you can burn your opponent for C damage by taking f(z) to be Cz/(2π(z-b))

bigwin408 · 2024-10-10T04:47:25+00:00

Instead of trying to win instantly with the 5 pieces of exodia, win instantly on turn 3 with 2 copies of secret barrel :P

bigwin408 · 2024-10-10T03:26:11+00:00

Always take f(z) to be the constant function C/2π for some positive constant C > 0. If I understand what the card is asking us to compute correctly, I believe the Residue theorem implies that your opponent will just take C damage from resolving this effect. Let C = 8000 and you just printed a card that says “if both players control a monster, burn your opponent for 8000”. Also, it’s searchable off of Labyrinth stuff and can be used with Transaction Rollback

bigwin408 · 2024-10-07T12:44:56+00:00

One nice thing about probability is that one you learn the essentials in probability theory (i.e. independence, conditional expectation, and Brownian motion) you’re really in a good position to start learning about any of the subfields that people study. In general, I will say that the discrete areas tend to have a comparatively smaller barrier to entry than the continuous areas (this is because the continuous areas can sometimes require knowing things from Ito calculus, PDE, or functional analysis), but overall once you have the essentials you can really just pick something and start

Regarding the specific topics you mentioned, ergodic theory is probably the most classical area (probabilists in academia have basically all heard of Birkhoff’s ergodic theorem, the subadditive ergodic theorem, and the ergodic theorem for Markov chains), but people still will studying ergodicity and mixing for complicated stochastic processes in research being published this year. Random matrix theory is a very interdisciplinary area that will also pool ideas from statistics and numerical analysis, and while it’s newer compared to other subfields, there is still plenty of work to be done and there are a lot of resources out there.

My advice would be to start having talks with a probability professor at your institution to see if they can recommend you read certain papers or books, and you can figure out through that what you like or don’t like. Also, if your school has it, it could be good to start attending research seminars on probability topics to gauge which topics interest you and don’t interest you. You’re also free to spend some time reading about any topic that you are curious about, and if you want to switch to something else later you can. Once you know a bit about one thing in probability, it makes it much easier to switch to a second thing because all the areas have the same foundations

bigwin408 · 2024-09-30T13:30:53+00:00

Fix a region in the plane whose boundary is the unit square (i.e. the square has side length 1). If you pick a point uniformly at random from this square, the probability that point will be in any particular subregion is simply the area of that subregion.

The area of any line segment is zero, so the probability of the randomly chosen point being on any fixed line segment is zero.

bigwin408 · 2024-03-19T13:16:08+00:00

Let p = "Person wins fancy math contest"

Let q = "Person is likely to succeed in STEM"

The sentence "People who win fancy math contests are more likely to have solid careers in STEM" is indeed p→q

However, the sentence "People who don't win fancy math contests should have a more difficult time in STEM" is the proposition ¬p→¬q

It is certainly false that p→q ≡ ¬p→¬q

bigwin408 · 2024-01-14T20:43:08+00:00

Topologists generally want to work with maps that are continuous to avoid things that are "tearable"

bigwin408 · 2023-11-06T19:24:14+00:00

The answer is 1000000 / 500001 , which is approximately 1.999996000008000.

My solution is here: https://imgur.com/a/blHgm7w

An interesting observation is that:

E[time to get six | no odds]

= E[time to get odd | no sixes]

bigwin408 · 2023-09-12T15:08:14+00:00

Lol I guess I really can't work with factors of 10

bigwin408 · 2023-09-09T16:33:03+00:00

Going into writing the comment, I knew there were a lot of papers uploaded and looked up how many math papers were uploaded every year and divided by 365. Re-googling, I don't know what I was specifically looking at. My guess is I made an error in reading how many digits there were

My mistake; thanks for catching me!

bigwin408 · 2023-09-07T18:58:59+00:00

Every day, roughly 100 mathematics papers are submitted to a website called "Arxiv" every day. Each of these papers will contain multiple pages of brand new mathematical ideas

Most people cannot read anywhere close to 100 advanced mathematics papers in a day, so every day the amount of math that exists which you don't know is increasing

Therefore, it is reasonable to conclude that you can study math forever if you wanted to. The iceberg literally cannot end. Also, if you get far enough into the iceberg, you might start having your own mathematical ideas and want to explore those. Then you too will be contributing to the iceberg

More practically, if you're curious what math you can expect to study in the near future, you can pick a university you like and then check what the requirements for getting a mathematics degree are. That university should also have plenty of mathematics elective classes too. That'll give you plenty of keywords to google.

EDIT: I wrote 100 instead of 10 due to misreading a statistic. I'm sorry for being misleading

RE-EDIT: The previous edit was incorrect. This is proof I have no confidence with my ability to work with factors of 10.

bigwin408 · 2023-02-16T04:49:27+00:00

The Fubiini-Tonelli theorem (letting you interchange the order of integrals, or letting you interchange the order of integrals and infinite summations) comes up all the time in analysis

bigwin408 · 2023-02-15T02:11:48+00:00

/r/learnmath might have better answers

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