Wikipedia:Reference desk/Archives/Mathematics/2013 February 6

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February 6

Newton–Pepys problem

Reading about Samuel Pepys, I came across this interesting exercise in probabilities. I've read the article linked in the subject heading and struggled to understand the maths. Please go easy on me. I understand the simple maths required to work out the probability of proposition A ("Six fair dice are tossed independently and at least one “6” appears.") but am struggling with B/C.

It seems that B and C are both calculated using binomial distribution, but that article is gibberish to me. Is there a simple way of explaining how the calculation given for (say) proposition B was arrived at - and perhaps less likely, is there a simple way to explain why the calculation works? NB Newton's intuitive reasoning, given at the bottom of the page, makes perfect sense. A genius mathematician who can explain things to us mathematical plebs - a role model, eh? --Dweller (talk) 11:18, 6 February 2013 (UTC)

For B (two or more 6s in twelve throws) you first calculate the probability of throwing no 6s. Each of the twelve throws must be between 1 and 5, so this probability is (5/6)^12, which is approximately 11.2%. Then you calculate the probaqbility of throwing just one 6. This is 12 x (1/6) x (5/6)^11 - the factor of 12 is included because the one 6 can be in any one of twelve places in the sequence of throws - which is 26.9%. The probability of throwing two or more 6s is then 1 - prob of no 6s - prob of one 6 = 100% - 11.2% - 26.9% = 61.9%. Same method works for C, except you also need to work out the probability of exactly two 6s and subtract this too. Gandalf61 (talk) 11:37, 6 February 2013 (UTC)

Thanks, that's quite digestible. --Dweller (talk) 11:50, 6 February 2013 (UTC)

Identity

$\lim _{n\rightarrow \infty }{\frac {\int _{0}^{1}x^{n}(1-x)^{n}f(x)dx}{\int _{0}^{1}x^{n}(1-x)^{n}dx}}=f(1/2)$. I can understand the intuition of this, but is there a proof?

Someone suggested expanding $f$ as a Taylor series.

$\lim _{n\rightarrow \infty }{\frac {\int _{0}^{1}x^{n}(1-x)^{n}\sum _{k=0}^{\infty }{\frac {(x-1/2)^{k}}{k!}}f^{(k)}(1/2)dx}{\int _{0}^{1}x^{n}(1-x)^{n}dx}}$

The integrand in the numerator is now a power series and can be easily integrated for known integer n (but remain a series), but I see no closed form for unknown n. What's the next step? --AnalysisAlgebra (talk) 19:30, 6 February 2013 (UTC)

Link to previous question

You didn't specify the conditions on f, I'll assume it's continuous at 1/2 and bounded.

Write the integral as the sum of an integral over a neighborhood around 1/2, and an integral over the rest of [0,1]. Bound the second integral using bounds on f and $x^{n}(1-x)^{n}$. In the neighborhood bound the distance between $f$ and $f(1/2)$ and use it to bound the distance between $\int _{1/2-\delta }^{1/2+\delta }x^{n}(1-x)^{n}f(x)dx$ and $\int _{1/2-\delta }^{1/2+\delta }x^{n}(1-x)^{n}f(1/2)dx$. Use these bounds and the triangle inequality to bound the distance between ${\frac {\int _{0}^{1}x^{n}(1-x)^{n}f(x)dx}{\int _{0}^{1}x^{n}(1-x)^{n}dx}}$ and $f(1/2)$. Tighten the bound as $n\to \infty$. -- Meni Rosenfeld (talk) 20:36, 6 February 2013 (UTC)

_______

Let $g(x;n):=x^{n}(1-x)^{n}$, and let $k_{n}$ = the integral of $g(x;n)$ from zero to one, and let $h(x;n):={\frac {g(x;n)}{k_{n}}}$. Hence the integral from zero to one of $h(x;n)$ equals one, so $h(x;n)$ (defined as zero outside the range zero to one) is a probability density function symmetric around 1/2. Using these, the OP's expression (upon cancelling out $k_{n}$ from numerator and denominator) becomes the limit of the integral from zero to one of $h(x;n)f(x)$, which is an expected value expression. My guess is that you can show that as n goes to infinity, $h(x;n)$ becomes zero everywhere except for a point mass of height one at x=1/2, which gives the desired result. (If my guess is wrong about the limit of h, then the desired result must be wrong.) Duoduoduo (talk) 23:02, 6 February 2013 (UTC)

Maclaurin's series, but where?

Dear Wikipedians:

I am confronted with the question "How many terms of the Maclaurin series for sin x are needed to guarantee an error less than 10^-5." But I feel that there is insufficient information given in the question for a single answer. Because depending on the specific value of x, either more or less terms are needed. As an example:

Maclaurin series for sin x is $\sum \limits _{n=0}^{\infty }{\frac {(-1)^{n}x^{2n+1}}{(2n+1)!}}$ where $x\in (-\infty ,\infty )$

If we truncate the series after the nth term, then the error E will be bounded by the size of the first omitted term $|E|\leq {\frac {1}{(2n+1)!}}x^{2n+1}$.

Since $x\in (-\infty ,\infty )$, if we let x = 1, then $|E|\leq {\frac {1}{(2n+1)!}}$ and we have

$|E|\leq {\frac {1}{7!}}={\frac {1}{5040}}=0.0001984>10^{-5}$, when n = 3;

$|E|\leq {\frac {1}{9!}}={\frac {1}{362800}}=0.000002755<10^{-5}$, when n = 4;

So we need at least 4 terms here.

But if we choose x = 2, then $|E|\leq {\frac {2^{2n+1}}{(2n+1)!}}$ and we have

$|E|\leq {\frac {2^{11}}{11!}}={\frac {8}{155925}}=0.00051306>10^{-5}$, when n = 5;

$|E|\leq {\frac {2^{13}}{13!}}={\frac {8}{6081075}}=0.000001315<10^{-5}$, when n = 6;

So we need at least 6 terms here.

And I would imagine that for higher values of x even more terms are needed for the Maclaurin series to converge as any polynomial would start to deviate more from the x axis around which sin x oscillates given sufficiently large x.

Therefore I feel that the question is not well-formed in the way it is being asked here and that x needs to be fixed around a particular value in order for the question to have a particular answer. Am I right with this line of reasoning?

Thanks,

76.75.148.30 (talk) 22:15, 6 February 2013 (UTC)

Maybe the question is intended to limit x to the range from 0 to 2$\pi$? If so, then work it out for the worst case, which is x=2$\pi$. Or maybe the intended range is $-\pi$ to $\pi$. Duoduoduo (talk) 23:36, 6 February 2013 (UTC)

Yes, the larger the argument, the more terms you will need. For any of the trig functions, the first step in numerical evaluation (whatever method is used (see for example CORDIC)), is to reduce the argument to the first quadrant [0,π/2], by using the periodicity and other symmetries of the function. So the question ought to specify 0 ≤ x ≤ π/2. --catslash (talk) 23:54, 6 February 2013 (UTC)

Thank you all for your response. That is what I have suspected all along. 69.158.77.73 (talk) 22:35, 7 February 2013 (UTC)