Wikipedia:Reference desk/Archives/Mathematics/2007 April 29

Welcome to the Wikipedia Mathematics Reference Desk Archives
The page you are currently viewing is an archive page. While you can leave answers for any questions shown below, please ask new questions on one of the current reference desk pages.

April 29

Maths Competition

I'm thinking about starting a maths competition. One which any one can participate and any methods can be used in the competition including astrology, palmistry or numerology.

The objective is to estimate the true probability of obtaining "Head" on a bias coin. The true probability of obtaining head with the bias coin can be expected range from 0.1 to 0.9 , in fact the coin toss is simulated by using a table of random numbers.

The procedure of the competition is as follows. The outcome of the coin toss would be annouced and each competitor would display their best estimate using all the knowledge they have obtained so far.

The full competition would ran for 40 rounds (thus involving 40 coin toss).

The winner would be the participant which has the lowest sum of error square.

A dummy run (of 3 rounds) is as follows: Assume that there are 3 competitors A, B and C.

The judge calls out "Head" and A displays "1.0", B displays "0.6", and C displays "0.5".

The judge calls out "Tail" and A displays "0.0", B displays "0.4", and C displays "0.4".

The judge calls out "Head" and A displays "0.7", B displays "0.7", and C displays "0.5".

The competition has ended and the judge announces the true probability of "Head" for the coin is 0.55 then the scores are calculated.

Coin Toss	Actual value	A's estimate	A's error square	B's estimate	B's error square	C's estimate	C's error square
Head	0.55	1.0	0.2025	0.6	0.0025	0.5	0.0025
Tail	0.55	0.0	0.3025	0.4	0.0225	0.4	0.0225
Head	0.55	0.7	0.0225	0.7	0.0225	0.5	0.0025
		Total	0.5275	Total	0.0475	Total	0.0275

The judge declares C the winner for having the smallest total of error square.

My question is this. If you want to win this competition, what is the best strategy and method for calculating the probability of "Head" in order to obtain the lowest possible error square score. Ohanian 01:33, 29 April 2007 (UTC)

How did the judge determine the value 0.55? Was it revealed in a dream? Was the coin from a lot guaranteed by the manufacturer to be 55-45 biased?  --Lambiam^Talk 05:33, 29 April 2007 (UTC)

A value for the probability of obtaining Head was chosen arbitrarily (beforehand) by the organizer of the competition. The value must be between 0.1 and 0.9 To simulate the coin toss, random numbers from a table of random numbers are used. If the random number (when converted to a form between 0 and 1) is less than (or equal to) the arbitrarily chosen value for Head then the Judge calls "Head" otherwise the Judge calls "Tail". Ohanian 05:51, 29 April 2007 (UTC)

Assuming all values between 0.1 and 0.9 are equally likely, the random variable for the bias, represented by the probability of heads in a single toss, has initially the uniform distribution with probability density function f(p) = 1.25 for 0.1 < p < 0.9, = 0 otherwise. (As usual for continuous distributions, we don't care about the density at boundary points.) The revealed outcome of each toss gives some information that allows the participants to refine the distribution, using Bayes' theorem, replacing each time f := f ^*, where the posterior's density f ^* is defined in terms of the density f of the prior by:

f ^*(p) = pf(p) / ∫ xf(x) dx           if the outcome was heads;

f ^*(p) = (1−p)f(p) / ∫ (1−x)f(x) dx if the outcome was tails.

(The integrals are definite, from x = 0.1 to x = 0.9.) The result will each time be a "censored" variant of the Beta distribution, restricted to 0.1 < p < 0.9. The best strategy is to give each time as the best guess the value g minimizing the expected value of the squared error, which is ∫ (x−g)²f(x) dx.  --Lambiam^Talk 07:16, 29 April 2007 (UTC)

I shall implement your advice using mathematica

(* Mathematica Input Format *)

f0[x_] := 5/4;

TEXT: Judge calls "Head"

f1[x_] := x*(f0[x]/Integrate[s*f0[s], {s, 0.1, 0.9}]);

Plot[f1[x], {x, 0.1, 0.9}];

errorsquare1[g_] := Integrate[(x - g)^2*f1[x],{x, 0.1, 0.9}]

OUT[4]= 0.41 - 1.213*g + 1.*g^2

Plot[errorsquare1[g], {g, 0.1, 0.9}];

FindMinimum[errorsquare1[g], {g, 0.501}]

OUT[6]= {0.0419556,{g->0.6067}}

TEXT: Judge calls "Tail"

f2[x_] := (1 - x)*(f1[x]/Integrate[(1-s)*f1[s],{s, 0.1, 0.9}]);

Plot[f2[x], {x, 0.1, 0.9}]

errorsquare2[g_] := Integrate[(x - g)^2*f2[x],{x, 0.1, 0.9}]

Plot[errorsquare2[g], {g, 0.1, 0.9}];

FindMinimum[errorsquare2[g], {g, 0.501}]

OUT[11]= {0.0271, {g -> 0.5}}

TEXT: Judge calls "Head"

f3[x_] := x*(f2[x]/Integrate[s*f2[s], {s, 0.1, 0.9}]);

Plot[f3[x], {x, 0.1, 0.9}];

errorsquare3[g_] := Integrate[(x - g)^2*f3[x],{x, 0.1, 0.9}]

Plot[errorsquare3[g], {g, 0.1, 0.9}];

FindMinimum[errorsquare3[g], {g, 0.501}]

OUT[16]= {0.0348,{g->0.5835}}

Coin Toss	Actual value	Lambiam's estimate	Lambiam's error square
Head	0.55	0.6067	0.0032
Tail	0.55	0.50	0.0025
Head	0.55	0.5835	0.0011
		Total	0.0068

Wow! I'm impressed by your method. It really gives a very low score (low is good) on the dummy run. 202.168.50.40 22:08, 29 April 2007 (UTC)

Okay, I did a dummy run too and I picked .55 three straight times so my error is even less. StatisticsMan 06:40, 1 May 2007 (UTC)

math property

I was wondering if there is a math property that allows you to move numbers around. ei 1+9*8+3. could you switch the 9 and the 8 around? —The preceding unsigned comment was added by 75.162.155.149 (talk) 02:03, 29 April 2007 (UTC).

First, in general you may substitute equals for equals. If you know that A = B, then also 123+456×A/789 = 123+456×B/789. Furthermore, both the operations of addition and of multiplication on numbers are commutative. If A and B are real or complex numbers (which includes integers), then A+B = B+A and A×B = B×A. So, e.g., 9×8 = 8×9. Because you may replace equals by equals, you may conclude now that also 1+9×8+3 = 1+8×9+3.  --Lambiam^Talk 05:24, 29 April 2007 (UTC)

Ah, but in 1×9+8×3 you cannot interchange the 9 and the 8 even though + is commutative. —Blotwell 18:48, 29 April 2007 (UTC)

To which we respond, of course, order of operations. In the absence of clarifying parentheses, multiplication must precede addition. Black Carrot 05:04, 30 April 2007 (UTC)

BODMAS. -- JackofOz 05:38, 30 April 2007 (UTC)

More qualifying examination preparation

Hey all, back again with another question from a previous qualifying examination that I can't quite seem to figure out. Here's the statement:

Let $f:[0,1]\rightarrow \mathbb {R}$ be continuous, with f(0) = 0 and $f'(0)$ exists. Show that $x^{-3/2}f(x)\in L^{p}[0,1]$ for all $1\leq p<2$.

Progress. Just so you don't think I'm a complete bum, let me tell you what I have right now. Since f is differentiable at 0, we know that $\lim _{x\to 0}{\frac {f(x)}{x}}$ exists. This suggests doing something like this:

$\int \left[x^{-3/2}f(x)\right]^{p}=\int \left[x^{-1/2}{\frac {f(x)}{x}}\right]^{p}=\int \left[x^{-p/2}\right]\cdot \left[{\frac {f(x)}{x}}\right]^{p}$,

and then note that since p < 2, $x^{-p/2}\in L^{1}[0,1]$. I feel like there should be some kind of use of Hoelder's inequality, but I can't quite figure out which two functions I would want to use. Hints would be greatly appreciated. Thanks! –King Bee ^{(τ • γ)} 17:45, 29 April 2007 (UTC)

Aren't you almost there already? Putting some mod signs in the above stuff, we have shown $\int |x^{-3/2}f(x)|^{p}=\int |x^{-p/2}|\cdot |f(x)/x|^{p}$ (sorry my markup is bad, too lazy to fix). But as you point out, $x^{-p/2}\in L^{1}[0,1]$, and since f(x)/x is continuous, [0,1] is compact and the limit at zero exists, $f(x)/x\in L^{\infty }[0,1]$. But then by the most trivial case of Holder, $\int |x^{-p/2}|\cdot |f(x)/x|^{p}\leq \|x^{-p/2}\|_{1}\|f(x)/x\|_{\infty }$, and we're done. Algebraist 18:19, 29 April 2007 (UTC)

Oh, jeez, I feel like such a freakin' idiot. Thanks for your help. –King Bee ^{(τ • γ)} 18:30, 29 April 2007 (UTC)

ps. what does qualifying exam signify here? Is this some americanism (or something) that always means qualifying for a graduate course, or a PhD, or something? Algebraist 18:24, 29 April 2007 (UTC)

I'm sorry for not explaining. At the university I attend as a graduate student, in order to get a PhD in mathematics, one needs to pass two written qualifying examinations (in addition to completing various other tasks). These written examinations are in four subjects, Algebra, Analysis, Numerical Analysis, and Applied Math. You choose your favorite two subjects from those four and then take a 3-hour written test at some point during your graduate career. (I've already taken and passed the algebra one, I'm now studying for Analysis.) Does this make sense? To answer your question, it probably is some kind of Americanism. –King Bee ^{(τ • γ)} 18:30, 29 April 2007 (UTC)

Perfect sense, except possibly for the lumping in of all pure maths into algebra and analysis. Does this involve a very wide definition of algebra, or does stuff like combinatorics just get ignored? Algebraist 18:37, 29 April 2007 (UTC)

At my university, combinatorics is completely ignored. The algebra exam is 10 questions, five of which are classical abstract algebra (groups, rings, etc.), and the other five are linear algebra (matrices, vector spaces, etc.). The analysis exam is also 10 questions, four of which are complex analysis, and the other six are real analysis. There used to be a wider array of subjects we could choose from, but they cut those exams out before I got here. And I should add, that it makes it really difficult for me, especially since my research is mainly in the realm of extremal graph theory and combinatorics! –King Bee ^{(τ • γ)} 18:43, 29 April 2007 (UTC)

Oh dear. Your university isn't in thrall to Bourbaki, is it? Algebraist 18:54, 29 April 2007 (UTC)

Negative; it is not. (Thank heavens.) –King Bee ^{(τ • γ)} 18:59, 29 April 2007 (UTC)

Error in Number Theory book?

Hi. I've been working through Number Theory by George Andrews, and doing each exercise, but I'm stuck on one where I think the text is in error. Either that, or I'm making a dumb mistake, because when I apply the hint given, it doesn't work. I've searched for a published errata for the book, but can't seem to find anything. The problem is #5 in section 8.2, on page 111.

The problem is connected with the previous one, which establishes an inequality following from the assumption that there is no prime number between $n$ and $2n$. In #5, we're asked to show that the inequality in #4 is impossible, given a couple of assumptions, which are both inequalities.

The equality that needs to be shown impossible (for sufficiently large n) is: ${2n \choose n}\leq \left(2n\right)^{\sqrt {2n}}R_{{\frac {2}{3}}n}$, where $R_{x}$ denotes the product of all primes not exceeding $x$.

The two inequalities that are supposed to contradict that one are $2^{n}\leq {2n \choose n}$ and $R_{x}\leq 4^{x}$.

The only sensible way I can see to combine those inequalities is just to line them up, and deduce that $2^{n}\leq \left(2n\right)^{\sqrt {2n}}4^{{\frac {2}{3}}n}$, but that doesn't seem to contradict anything, since you've got $2^{n}$ on the left, and something bigger than $2^{{\frac {4}{3}}n}$ on the right.

Am I missing something, or is there a typo in the book, or what? Has somebody encountered this problem before, and made sense of it? -GTBacchus^(talk) 19:26, 29 April 2007 (UTC)

I don't have the book, but given the fact that ${\tbinom {2n}{n}}$ is asymptotically equal to $4^{n}(\pi n)^{-{\frac {1}{2}}}\,$, the lower bound $2^{n}\,\!$ is ridiculously weak.  --Lambiam^Talk 06:26, 30 April 2007 (UTC)

Oh, it's weak. The only reason it's in the book is because it's a step in establishing the first half of Tchebyshev's result that ${\frac {\ln 2}{4}}\cdot {\frac {x}{\ln x}}\leq \pi \left(x\right)\leq 30\left(\ln 2\right){\frac {x}{\ln x}}$, which is weaker than the Prime Number Theorem. It's not a typo, though because it does work for establishing that result; it just doesn't seem adequate to solve the exercise I'm working on. :/ -GTBacchus^(talk) 07:27, 30 April 2007 (UTC)