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Comment by daveFNbuck | Hacker News Reader

I got a B in probability because I didn't write a proof of the central limit theorem on the allowed cheat sheet for the final exam. So of course it's the first thing I looked for on this one. It's not there.

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Aren't most proofs of the CLT pretty extensive?

There are a few short ones going from area integral comparisons.

The easiest is to look at characteristic functions and cumulants; for a random variable T with PDF p(t) we say T ~ p and define

φ_T[f] = ∫ dt p(t) exp(-2πift) = ⟨ exp(-2πifT) ⟩

If two variables X ~ r and Y ~ s are independent then you can prove [from ⟨f(X) g(Y)⟩ = ⟨f(X)⟩ ⟨g(Y)⟩ or X+Y ~ q where q(z) = ∫ dx r(x) s(z — x)] that their sum has a characteristic function

φ_{X+Y} = φ_X + φ_Y

And therefore the “sample mean” M of n IID variables is itself a random variable with characteristic function

φ_M[f] = ( φ[f/n] )^n.

So we find that

log φ_M[f] = n log φ[f/n] ≈ 0 + i a f – b f²/n + O(f³/n²).

These terms [a, b] from expanding the log of the characteristic function constitute the cumulant expansion and for large n the other terms shrink to zero, so that the characteristic function is to first order in 1/n a Gaussian.

The characteristic function was a Fourier transform of a PDF, so an inverse Fourier transform gets it back:

p(t) = ∫ df φ_T[f] exp(2πift)

But the Fourier transform of a Gaussian is just a Gaussian.

Surprise you can use those character like “∫ φ π”. Thought you need to say pig etc.

A B is like 80% or so, judging from A being the best iirc? I'm not familiar with alphanumeric grading systems.

Things can get a bit customized at either end of the spectrum, but generally each letter covers 15%, with A..D covering the span 100% to 40%.

So a B would be 70 to 85.

Shouldn't that sort of grading be on a bell curve? That would seem to be the fairest.

Many smaller (<60 students) math classes, in my experience, have bimodal distributions of scores, so a bell curve by definition simply doesn't make sense. In addition, there's been a move to standards-based or mastery-based grading, that is, grading according to what you know of the material rather than how you compare to your neighbor. This allows comparisons over time and consistency with regard to subsequent classes -- if you have a C as a prerequisite for the next class, then a C should indicate the same mastery of material rather than the same relative position in the class.

Based on my memory of what the teacher said in my statistics class in school, exam results in Scotland are normalized as z-scores vs that year's population. At least they were a zillion years ago.

Let's say that most math majors take this course in the first semester of their sophomore year. Then the bell curve grading gives you a much lower grade for identical work if you take the class in the first semester vs second semester. You also get a lower grade for going to a better school. You even get a lower grade for helping your classmates.

This makes the grades unfair and not useful for judging mastery of the subject. It only makes sense if the goal of a course is to beat the other students. For most courses, the goal should be learning.

Is this a new thing? I graduated highschool in 2001 and each grade letter was 10% and equated to a grade point.

    90%+ => A (4.0)
    80%+ => B (3.0)
    70%+ => C (2.0)
    60%+ => D or F depending (1.0 or 0.0)

At university, we only used grade points and they were very, very similar in scale.

It was the grading system in Ireland when I graduated secondary school. However, it looks like the meanings of the letter bands varies dramatically between countries.

70 - 85 for a B?! I wish I had that deal.

At my university the letter grades were

    A+  99.00 - 100.00%
    A   93.00 -  98.99%
    A-  90.00 -  92.99%
    B+  87.00 -  89.99%
    B   83.00 -  86.99%
    B-  80.00 -  82.99%
    C+  77.00 -  79.99%
    C   73.00 -  76.99%

Many places in the US, each grade covers 10%, so As (with plus or minus) go between 90 and 100, etc, with D from 60-70 and F below 60.

It's typically the 80-90% range. It's also typically divided into thirds for B-/B/B+. I actually got a B+, so it would have been closer to 90%.

Question asking for the central theorem proof in a probability class happened to me too. Not answering this one gave me an A, though.

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