Wikipedia:Reference desk archive/Mathematics/2006 August 6

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[edit] Asymptotic Behavior

For the product f(n) = \prod_{p=3}^{p_n} \frac{p}{p-1}, given a prime pn, I can tell that it's quite a bit less than linear to pn, since it has to be less than primorial(pn)/primorial(pn-1), so it's less than nln(n). That means, I believe, that it's a low polynomial to n, since n>sqrt(n)>ln(n), meaning n3/2>f(n). It seems like it's at most linear to n, though, perhaps even logarithmic. Can anyone make that more definite? Black Carrot 00:30, 6 August 2006 (UTC)

If it helps, the numerators in the sequence f(2), f(3), ... are (sequence A060753 in OEIS) and the denominators are 2 × (sequence A038110 in OEIS). Fredrik Johansson 01:18, 6 August 2006 (UTC)
I think you want Mertens' theorems. JoshuaZ 15:03, 7 August 2006 (UTC)

[edit] LaTeX recurring symbol

I've read through a few LaTeX tutorials, and while I feel sure this must be possible I've found none that could help - I am writing a document where certain symbols are going to show up fairly regularly, and I'd like to define a macro or whatever that will make it easier to put them in. For example, when I want to put in a Christoffel symbol, instead of having to type \Gamma^{\alpha}_{\phantom{\alpha}\beta\gamma} or similar, I would type in \chr2{\alpha,\beta,\gamma} or something. How do I do it? Confusing Manifestation 07:48, 6 August 2006 (UTC)

If you want to do exactly that, put this at the top of your document:
\def\chr2#1#2#3{\Gamma^{#1}_{\phantom{#1}#2#3}}
Then you can use it by typing
\chr2{\alpha}{\beta}{\gamma}
I should admit that I am a plain TeX user. I think there's a more LaTeXish way to define macros, but this way will work. —Bkell (talk) 07:54, 6 August 2006 (UTC)
Yes, LaTeX has a nicer way. This page has, coincidentally, an example involving Christoffel symbols. --KSmrqT 08:54, 6 August 2006 (UTC)
That is incredibly awesome and extremely helpful. And impressively fast! Thanks! Confusing Manifestation 09:10, 6 August 2006 (UTC)
Heh, "nicer" depends on your point of view. I've used both LaTeX and plain TeX methods, and to me it seems that the plain TeX ways are almost always more elegant. ;-) —Bkell (talk) 09:11, 6 August 2006 (UTC)
Hear, hear! Dysprosia 22:24, 6 August 2006 (UTC)

[edit] statistics

Hello, I have a final exam of statistics tomorrow, and I need at least 80 points out of 100 to be able to pass this course. Though I know the “do your own homework” rule, I’m posting a question that I could not figure out on my own, and with my apologies... If anyone can answer it or at least tell me which rule to apply, I’ll be most grateful... Question: “A firm is manufacturing ball bearings for a precision tool maker who specifies that the individual balls are to be graded such that those he receives are 0.500+-0.005 in. diameter. Several samples of 100 balls have been taken, and the result are 0.5000 in. mean and 0.00312 in. standard deviation. What percentage of the balls will be rejected with this criteria?” thanks

Putting μ = 0.5 and σ = 0.00312, the requirement on the random variable X representing the diameter in inches translates into μ −1.6026σ < X < μ +1.6026σ. Assuming normal distribution for X, the probability of non-rejection is then Φ(1.6026) − Φ(−1.6026) = 2Φ(1.6026) − 1. So the probability of rejection is 2(1 − Φ(1.6026)). Using that Φ(1.6026) is about 0.9455, we find a rejection percentage of 10.9 %.  --LambiamTalk 18:48, 6 August 2006 (UTC)

[edit] What is the name of this game?

Hi. 3 years ago, while studying mathematics, a friend told me of a mathematical game. Unfortunately I've forgot quite a few details, including the name, so I can't look it up.

The game is for one person, and is divided into rounds, and at round n, the outcome is f(n). I remember f to be strictly increasing, even exponential. We have f(1)=1. Each round, a coin is flipped. If heads, the game ends, and the person receives f(n), and if tails, the game continues.

In the analysis one wishes to calculate the estimated outcome of participating in this game. The estimated outcome turns out to infinitely large, which is very surprising.

The analysis goes like this: If we restrict the game to k rounds, the estimated return is the sum of the first k harmonic numbers. Since that sum is divergent, we get an infinite return - but in practice the harmonic numbers grows very slowly. Further, f(n) grows extremely quickly (to counteract the exponentially bad odds that we reach a given round), so the money involved quickly becomes unreasonably large.

Thanks in advance Søren

It sounds like you may be looking for the St. Petersburg paradox. JoshuaZ 21:35, 6 August 2006 (UTC)
Indeed, that's the one. Thanks. Søren
For a less verbose treatment, see Martingale (betting system). --LambiamTalk 07:12, 7 August 2006 (UTC)