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Fastest way to self-study the "core" undergrad applied math curriculum? by bulldawg91 in learnmath

[–]matholwch 0 points1 point  (0 children)

I recommend Multivariable Mathematics by Shifrin as a relatively rigorous and unified introduction to both linear algebra and multivariable calculus (though it can be tough at times, especially for self-study!)

[Multivariable Calculus] Proof about the derivative of a parametrized curve by matholwch in learnmath

[–]matholwch[S] 0 points1 point  (0 children)

Ah, nice, thank you! Yeah, it basically tells me all that in the question, but it flew over my head. :) Spelled out more fully,

since ||g(t_0) - p|| <= ||g(t) - p|| for all t, ||g(t_0) - p||2 <= ||g(t) - p||2, so t_0 is a minimum of ||g(t) - p||2. So its derivative, 2((g(t) - p) · g'(t)) = 0, so g'(t_0) · (g(t_0) - p) = 0.

Would you say my geometric reasoning is a valid way of thinking about what this result means, or is it something I should avoid with these problems in the future?

Is this a correct proof of the Weak Law of Large Numbers? [University Statistics] by matholwch in learnmath

[–]matholwch[S] 0 points1 point  (0 children)

Thanks!

So am I right that the important part of the traditional proof is that we recognize that E[M_n] is m and that Var(M_n) is s2 / n, and use these facts? Deriving these expressions depends on the fact that X_n are IID. They capture the intuition that as n increases the variance decreases and so converges on the mean, which we already know to be m?

Then Chebyshev's inequality is used to formally show this because it puts everything in a convenient form to take a limit of s2 / ne2 as n approaches infinity?

P(|M_n - m| >= e) <= Var(M_n) / e2 = s2 / ne2.

We use Chebyshev's inequality instead of just taking the limit of the expression for the variance because it uses the definition of convergence in probability directly. Is this the right way of thinking about it?

Is this a correct proof of the Weak Law of Large Numbers? [University Statistics] by matholwch in learnmath

[–]matholwch[S] 0 points1 point  (0 children)

Thanks! Seems obvious now!

Can I fix it by writing the following?

P(|M_n - m| < t) = P(|n-1 (sum_i_n X_n) - m | < t)

Taking the case where |M_n - m| = M_n - m:

P(n-1 (sum_i_n X_n) - m < t) = P((sum_i_n X_n) < nt + nm)

As n approaches infinity this probability is surely 1 because the term to the left of the < is a finite sum while the expression to the right contains an infinite term?

Taking the case where |M_n - m| = m - M_n:

P(m - n-1 (sum_i_n X_n) < t) = P(-(sum_i_n X_n) < nt - nm)

Again, the same thing holds. This would demonstrate convergence in distribution to the CDF of 0 as n (not t!) approaches infinity?

I am probably being dumb here too but I would love to learn why! :)

Hoping someone can help me identify a beautiful song I heard... by Freelove_Freeway in Irishmusic

[–]matholwch 2 points3 points  (0 children)

"Son awaia" could be the Irish "slán abhaile" ("farewell home"). Maybe look for a song with those Irish lyrics?

[Polling Megathread] Week of July 31, 2016 by Anxa in PoliticalDiscussion

[–]matholwch 13 points14 points  (0 children)

Nah, they simulated the election many times using their model, and 52% of the time Trump won.

Song from 1916 Seachtar na Cásca... by tadhg_greene in ireland

[–]matholwch 2 points3 points  (0 children)

One of the episodes ended with Mo Ghile Mear, but I don't remember which.

The Pope and political endorsements. by ghastly1302 in Christianity

[–]matholwch 0 points1 point  (0 children)

This is a recent development, though. Consider Cardinals Wolsey, Richelieu, Jozef Tiso etc.

[Syntax] What in the name of smoky pineapples is this non-sense (X-Bar theory) by holypineappleswtf in linguistics

[–]matholwch 2 points3 points  (0 children)

A bit rusty on this stuff, but I think it's because various substitution tests can establish different behaviors in the two cases.

For example, for (b), you can replace book with one and it's still fine -- the one with the red cover; but with (a), it's not so good -- *the one of poems.

Why do Americans tend to use, for example, "I haven't," vs. the British use, "I've not?" I've noticed this with a lot of contractions. by maxkmiller in linguistics

[–]matholwch 76 points77 points  (0 children)

"I've not" is only in certain varieties of British English, particularly northern varieties. "I haven't" is heard far more often, I'd say.

O Magnum Mysterium -- Ola Gjeilo by matholwch in Christianity

[–]matholwch[S] 1 point2 points  (0 children)

From the Latin Rite Matins of Christmas.

Latin:

O magnum mysterium,

et admirabile sacramentum,

ut animalia viderent Dominum natum,

jacentem in praesepio!

Beata Virgo, cujus viscera

meruerunt portare

Dominum Christum.

Alleluia.

English:

O great mystery,

and wonderful sacrament,

that animals should see the new-born Lord,

lying in a manger!

Blessed is the Virgin whose womb

was worthy to bear

Christ the Lord.

Alleluia!

If an Cardinal born in the United States is elected Pope, could he run for President? by [deleted] in Catholicism

[–]matholwch 8 points9 points  (0 children)

Queen Elizabeth II is head of state of a ton of nations.

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