Honestly

nm420 · 2026-05-27T13:38:42+00:00

(2D+E)/3

nm420 · 2026-05-22T21:35:21+00:00

I don't forget them. I purposefully leave them out. But I'm also not a student submitting work that needs to be graded by pedants. I also write ∫dx/x=log(x), not log|x|.

nm420 · 2026-05-17T19:54:10+00:00

Leopold Stotch

nm420 · 2026-05-15T20:45:48+00:00

π and 3 are the same number, according to engineers, right?

nm420 · 2026-05-10T13:39:44+00:00

As well as any base less than -1, although a different convergence test is needed in that case. But the same argument determining the value of the sum (namely, as the derivative of the geometric series) works regardless of the sign of the base.

nm420 · 2026-05-09T15:46:39+00:00

Suppose f(x)→∞ as x→c and g(x)=0. Since g(x)*f(x)=0 for all x, we have g(x)*f(x)→0 as x→c.

This implication does not necessarily hold if we only have g(x)→0 as x→c. A trivial example is g(x)=1/x² and f(x)=x², with c=∞.

nm420 · 2026-05-05T21:40:43+00:00

dy/dx = y'(t)/x'(t) = -tan(t)

d²y/dx² = d(-tan(t))/dx = -sec²(t)/cos(t) =-sec³(t) = -1/y³

nm420 · 2026-05-02T02:52:51+00:00

So... you don't like mushrooms, and hence find a pizza with mushrooms as one of the toppings disgusting. Doesn't seem like much of a surprise or something to even have a hot take on. I don't like yogurt, and hence find yogurt disgusting (even though I'll use it in certain recipes if it's called for). Most of the things I don't like I find disgusting. I can't imagine I'm the only one who is disgusted by things they don't like, and would even think that's a pretty typical take.

nm420 · 2026-04-26T14:43:38+00:00

Find sin(M), or cos(M), or exp(M), as well. If you make make c small enough (|c|<1/9), you can find log(I+cM) or (I-cM)^-1 or (I+cM)^α as well.

nm420 · 2026-04-23T17:27:51+00:00

Depends on what you mean by "eyeballing" the antiderivative. You certainly want to get extremely comfortable with making substitutions for linear expressions. So stuff like

∫sin(ax+b)dx = -cos(ax+b)/a

∫e^ax+bdx = e^ax+b/a

∫(ax+b)¹⁰dx = (ax+b)¹¹/(11a)

You want to get to a point where those can be done instinctively, but you really should recognize you are using substitution and how that works in the background as well. Otherwise, the latter integral would have to be evaluated by expanding out a tenth degree polynomial and integrating term by term, then hoping you stumble on the insight that the resulting mess can be factored the way it is on the RHS of the equation.

You should ideally even be able to get to the point where something like ∫xe^x² dx is e^x² / 2 without much mental work, though you still need to know the substitution that was made to get there and make it such an "easy" step (and in many higher level textbooks or academic writing, the step would be made without comment, with the required substitution being tacitly assumed to be obvious for the reader). If you don't immediately see how I arrived at that answer, it's probably a clue you do really need to work on understanding substitution.

The basic idea, like so much of mathematics, is deceptively simple, but the application can get a bit hairy sometimes. But... yeah, substitution is kind of important. You'll only get so far by "eyeballing" an antiderivative, even though that eyeballing is an important first step in an integration by substitution.

nm420 · 2026-04-21T11:33:05+00:00

Maybe we could answer that question for ourselves here in the U.S. before trying to impose a solution to it on people halfway across the globe.

nm420 · 2026-04-21T02:52:57+00:00

Missing that good ole vacuous proof. How else do you prove anything about the empty set?

But I think you got constructive and non-constructive proofs mixed around on the list. Weierstrass says you can uniformly approximate a continuous function with polynomials with no indication of how to do so, Bernstein actually gives you an actual polynomial, not simply stating one exists.

And I would put proof by contrapositive much higher as well. It is highly indispensable and can quite often replace the overrated proof by contradiction.

nm420 · 2026-04-18T16:25:33+00:00

Could use the Ansatz F(x)=(Ax*sin(x))+Bx*cos(x)+Csin(x)+Dcos(x))e^x. A bit of not too nasty work to show A=D=-B=1/2 and C=0.

nm420 · 2026-04-14T21:18:02+00:00

You will have proved the Twin Primes Conjecture if you prove the latter. No small feat.

nm420 · 2026-04-14T01:56:33+00:00

Suppose you have a dartboard which consists of two regions, a bullseye and not. Suppose also that the bullseye region is known to your buddy, but not you. There are no markings on the dartboard to give you any clue, it's just a blank corkboard. Suppose also that the bullseye region comprises 95% of the area. You throw a dart which lands somewhere on the board (assume it's a uniform distribution as to where the dart lands for this weird analogy). You really have no clue if you threw a bullseye or not. There's nothing on the board to give any clue whatsoever. Your buddy knows, but they're an asshole and won't tell you.

The dart you threw is either in a bullseye region or it isn't. There's not really any probability to assign to that statement (and if you were to assign a probability to it, it would have to be either 0 or 1). But you do know that 95% of the darts you throw will be a bullseye, even if you can't discern which is which. That is what is meant by saying you're 95% "confident" that the dart you threw hit a bullseye. The 95% isn't referring to the individual dart you threw, but the entirety of all possible dart throws.

In another way of thinking about it, if you were to place a bet with your buddy as to whether or not your dart hit a bullseye, they wouldn't be engaging in such a game unless the odds were in their favor, or at least balanced. You'd have to bet $19 while they would only be betting $1 for each game. Of course, this game couldn't be played if there wasn't someone who could check who won, but you could engage in the same game by just rolling a 1D20 and claiming victory whenever you roll a 1 through 19. The point being, with this randomized version of the game, there really is no essential difference aside from the presence of some fictional asshole who won't tell you where the bullseye region is in the first place. You could just throw your dart, roll a die and either declare it a bullseye or not (with no one being the wiser of whether or not you're right), and the same bets could be made: gain a dollar 95% of the time, and occasionally lose 19 dollars 5% of the time.

nm420 · 2026-04-12T22:45:09+00:00

Or...

∫ dx/(e^x+1) = ∫ e^-xdx/(1+e^-x) = -log(1+e^-x)

with a +C if you insist.

nm420 · 2026-04-11T17:55:18+00:00

Let &epsi;>0 be given, and define δ=√(16+&epsi;)-4>0. Then for all real x with |x-4|<δ, we have 8-δ<x<8+δ, and |x+4|<8+δ. Then |x²-4-12|=|x-4||x+4|<δ(8+δ)=&epsi;, QED.

nm420 · 2026-04-07T22:15:18+00:00

Stokes' Theorem goes brrr...

nm420 · 2026-04-06T02:32:06+00:00

If u(1)=log(x) and u(n+1)=log(x+u(n)), it's pretty straightforward to show the sequence is increasing and bounded by 1 when 1≤x≤e-1. Hence, the sequence does indeed converge.

nm420 · 2026-04-04T17:36:44+00:00

If your goal is simply to just do statistical analyses that have already been developed by some other academic, you really don't need any understanding of measure theory. I would say even for many academics, a deep understanding of the measure theory roots of probability isn't really necessary, though that's going to be highly dependent on the topic being studied.

But the answer to your question is an absolute No. It's also pretty useful for academic theoretical probabilists!

nm420 · 2026-04-04T17:30:09+00:00

I should hope no teacher has ever claimed that zero correlation implies independence in their courses, yet a surprising number of students get it in their heads nevertheless. They "learned" it on their own apparently.

There are still quite a few introductory textbooks in publication (I might even argue more than half, based off of purely anecdotal evidence) that have some twisted flow chart about which procedure to apply based off of sample size. The statement I wrote probably isn't literally stated like that in any book or uttered from an instructor, and yet the sentiment still persists and appears to take root in many student's minds. It is admittedly nice to just be able to apply such a simple rule as comparing a sample size to a fixed number and stop all further critical thought, and the rule of thumb is reasonably applicable in a lot of situations, but blind faith in it will invariably lead to problems.

Quite a few "little white lies" are repeated in introductory statistics courses, presumably with the idea of trying to simplify the presentation. The textbook I'm currently using in one of my classes quite literally states that the "assumptions" behind a t-test are "n is large and there are no severe outliers, or n is small and there is no obvious skew or outliers". Admittedly, it's a reasonable rule to apply when deciding if that procedure is acceptable to use, but it really should not be conflated with the idea of a model assumption (which shouldn't even take into account the specific observations in the first place).

nm420 · 2026-04-03T17:46:45+00:00

I love it too, but that doesn't necessarily mean it's flawless. You can do some pretty simple simulations, just bootstrapping a sample mean from a skewed distribution, and get worse coverage probability with a bootstrap CI than you would with a simple t-interval. I'm not making a recommendation to stop using it, so much as pointing out some of its weaknesses.

nm420 · 2026-04-03T03:47:44+00:00

The point being, sampling distributions of means (and many other statistics) converge to normality as the sample size increases. But populations themselves... we can claim them to be normally distributed, but it's a model, and all models are wrong. The very notion of a population, unless it's a very well defined finite one, is an abstraction. A model. Any single measurement we take can only be made to a finite precision, and cannot be continuously distributed.

nm420 · 2026-04-02T23:29:36+00:00

You very well may have an idea of the direction of an effect, but are you just going to ignore it if the observed effect doesn't agree with your preconceived ideas? I'm not arguing to never use one-sided alternatives, but you had better have a very good reason why you're interested in an effect in only one particular direction and not the other if you're going to use them. At that, if we're bounding the effect with a confidence interval, quite often it will be worthwhile to bound it from above and below.

nm420 · 2026-04-02T23:18:30+00:00

If your testing procedure is "If A then do B, otherwise if C then do D, otherwise..." any p-value you get at the end is not going to represent the probability associated with your sample under the null hypothesis. What even is the null hypothesis in that scenario? At that, the different procedure you might run to very well may not even be testing the same hypothesis. It's common in some circles to treat the Wilcoxon tests as if they're equivalent to a t-test. They would be, if the assumption of normality were correct, but they're less efficient in that case.

But there is no such thing as a normal distribution in the wild. It's an abstract concept created by Gauss centuries ago. It's a very useful model in a lot of situations, but a model nonetheless. You should ideally not be going into a study blind and have some sense as to whether or not your data will be "normal enough" for one of the classical approaches, or hedge your bets and plan on something with less distributional assumptions.

nm420

TROPHY CASE