Wikipedia:Reference desk/Archives/Mathematics/2010 October 24

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October 24

Differentiate a function

How would I compute ${\frac {d}{dx}}(x!)$, assuming that non-integer values of n! exist and can be computed using the gamma function/Stirling's approximation. 24.92.78.167 (talk) 02:56, 24 October 2010 (UTC)

Technically, non-integer values of n! do not exist, by definition. If we were to use n! as a shorthand for $\Gamma (n+1)$, which is equal for integer values of n, then, according to the page on the gamma function,

${\frac {d}{dx}}(x!)={\frac {d}{dx}}\Gamma (x+1)=x!\cdot \left(-\gamma +\sum _{k=1}^{x}{\frac {1}{k}}\right),$

where $\gamma$ is the Euler–Mascheroni constant. According to Stirling's approximation,

$x!\sim {\sqrt {2\pi x}}\left({\frac {x}{e}}\right)^{x}.$

${\frac {d}{dx}}({\sqrt {2\pi x}}\left({\frac {x}{e}}\right)^{x})={\sqrt {\frac {\pi }{2}}}e^{-x}x^{x-1/2}(2x\log(x)+1)$

Since Stirling's approximation is different then the gamma function and they are not even equal at integer values, it is no surprise that this is a very different formula. 76.67.73.90 (talk) 03:31, 24 October 2010 (UTC)

The formulas aren't that different. ${\sqrt {\frac {\pi }{2}}}e^{-x}x^{x-1/2}(2x\log(x)+1)={\sqrt {2\pi x}}e^{-x}x^{x}(\log x+1/2x)\approx x!(\log x+1/2x)$. It is well known that $-\gamma +\sum _{k=1}^{x}{\frac {1}{k}}\approx \log x$ and slightly less well known that it is $\approx \log x+1/2x$. -- Meni Rosenfeld (talk) 06:55, 24 October 2010 (UTC)

"76.67.73.90", you wrote:

${\frac {d}{dx}}(x!)={\frac {d}{dx}}\Gamma (x+1)=x!\cdot \left(-\gamma +\sum _{k=1}^{m}{\frac {1}{k}}\right),$

Can you explain what you mean by m? What is m? Michael Hardy (talk) 04:43, 24 October 2010 (UTC)

Looking at the linked page on the Gamma function, they meant x instead of m. It should be noted x is an integer in that formula (...since otherwise x! isn't defined). 67.158.43.41 (talk) 04:50, 24 October 2010 (UTC)

Remarks. It turns out that what is easier to write is the logarithmic derivative of $\Gamma (x)$, that is $\Psi (x):={\frac {\Gamma '(x)}{\Gamma (x)}},$ rather than $\Gamma '(x)$ itself. Actually, the construction (or one construction, at least) of the $\Gamma$ function starts from the construction of the digamma function $\Psi (x),$ as a solution of a simple linear functional equation. For this reason, the simplest formula for the derivative of $\Gamma (x)$ is $\Gamma '(x)=\Psi (x)\Gamma (x)$. As to the series expansion for $\Psi (x),$ you can find it here; the formulae given above by anonymous users are wrong or nonsensical for non-integer x. --pma 13:43, 27 October 2010 (UTC)

Finite subgroups of the multiplication group of the unit quaternions

I want a list of all finite subgroups of the multiplication group of the unit quaternions. --84.61.153.119 (talk) 08:30, 24 October 2010 (UTC)

You really should do your own homework. At least show some effort. If you don't know what a quaternion is, you may find the page quaternion group helpful. Jkasd 08:50, 24 October 2010 (UTC)

Also the list of small groups may be useful. --JohnBlackburne^words_deeds 11:37, 24 October 2010 (UTC)

I want a list of all finite subgroups of the group O(4). --84.61.153.119 (talk) 14:37, 24 October 2010 (UTC)

Does the position of the median change if a plot is distorted like this?

Let's say I have a large number of values, creating a one dimensional plot, and I figure out what the median is and in what places on the graph it occurs. Then I want to change the values so that the highest and lowest values remain the same, while the current median value is changed, and all values in-between are "scaled" to allow this to happen. If I calculate the median again, will I always find it in the same places on the graph as before? I think it should be but I lack the knowledge to verify this. Thanks. (Sorry, I can't express this in any other way than with words, but I can try to rephrase it if it's hard to understand) Obiha (talk) 11:12, 24 October 2010 (UTC)

The median for a finite number of discrete values is simply the middle value: more precisely if there are an odd number of values, n say, it's the (n + 1)⁄2 value, if there are an even number it's midway between the n⁄2 and n⁄2 + 1 values. To change the median you need to change just these values, but keeping them at their positions which may mean moving other values too. E.g. if the values are

1  5  9 13 17 21 25 29 33

the median is 17, the 5th element. To change the median to 21 you could change the numbers like so

1  6 11 16 21 23 25 28 31

The first and last numbers are unchanged but everything else has changed.--JohnBlackburne^words_deeds 11:47, 24 October 2010 (UTC)

What the OP asked is: He has an unsorted sequence of values, say

1 4 3 9 4 5 4

He finds their median 4, and then notes that it appears is position 2, 5 and 7. Then he applies a transformation which changes the median - I don't know if he means specifically a linear scaling, but for now assume that the transformed values are:

1 6 4 9 6 7 6

What he wants to know is - if he takes these new values, find their median and note where it appears, will he also get 2, 5 and 7? -- Meni Rosenfeld (talk) 12:04, 24 October 2010 (UTC)

[ec] Yes, if either there are an odd number of values or the middle two values are equal (otherwise it could depend on your definition of median, and your method for finding it). Let $x_{1},\ldots ,x_{n}$ be the values, f be the transformation you apply to each, $y_{i}=f(x_{i})$ be the transformed values, and ${\tilde {x}},{\tilde {y}}$ be the medians of x and y. A median is characterized by the fact that at most half the values are less than it and at most half the values are greater than it. So if f merely satisfies $a>{\tilde {x}}\Rightarrow f(a)>f({\tilde {x}})$ and $a<{\tilde {x}}\Rightarrow f(a)<f({\tilde {x}})$, it will follow that $f({\tilde {x}})$ is the median of y. From these conditions it will also follow that $a={\tilde {x}}\iff f(a)=f({\tilde {x}})$ which means that $x_{i}={\tilde {x}}\iff y_{i}={\tilde {y}}$, so the median will appear in the same places for both sets. -- Meni Rosenfeld (talk) 11:59, 24 October 2010 (UTC)

Venn Diagram Word Problems

I recently undertook a logic and problem solving course at a local community college as part of an IT diploma. We just commenced Venn diagrams and I seem to be having a lot of difficulty with them. Can anyone recommend a resource or site which contains questions on which to practice? I have used this site so far http://www.math.tamu.edu/~kahlig/venn/venn.html and I am looking for more?24.89.210.71 (talk) 12:16, 24 October 2010 (UTC)

Problem

Sir i have on problem plz solove it.

If:

  7-3=124
  6+3=279
  5-2=63
  11+2=2613

then 15-3=? —Preceding unsigned comment added by Apsirkeel (talk • contribs) 12:58, 24 October 2010 (UTC)

I'll give you a hint:

7-3=12|4
6+3=27|9
5-2=6|3
11+2=26|13
15-3=??|??

-- Meni Rosenfeld (talk) 13:11, 24 October 2010 (UTC)

Choosing a pineapple at a supermarket

The pineapples, for example, at a supermarket are all the same price but vary in quality according to a Gaussian distribution. If I choose three pineapples at random from the pile, and then select the best of those three, what is the probability that the selected pineapple is within the top ten percent of quality of the whole pile?

How many pineapples do I need to compare to have at least a 50% chance of being in the top x percent of quality? Not a homework question, but a practical problem. Thanks 92.15.31.47 (talk) 16:57, 24 October 2010 (UTC)

ditto for wives, guys. how many past candidates should you first reject in order to have a sample base, and afterward pick the first one better than all of these? Assume that although normally distributed, the "value" is hidden to you, and you have no absolute way to compare candidates, but instead can only do a comparison function on candidates you have already rejected? You don't want to waste your time, so we're looking at the least number to have some high confidence (as with the number above) 84.153.247.182 (talk) 17:48, 24 October 2010 (UTC)

See Secretary problem. —Bkell (talk) 18:13, 24 October 2010 (UTC)

The answer to the first is straightforward, and the distribution doesn't matter. The chance of picking 1 that's in the top 10% is 0.1. The chance it's not is 0.9. If you pick three the chance all three are not in the top 10% is 0.9³ = 0.729. The probability at least one is in the top 10%, so that the best is in the top 10%, is therefore 0.271.

If you want to choose n so there's at least a 50% chance that the best of those is in the top 10% then you need n = 7, as 0.9⁷ = 0.478, so there's about a 52% change one of the seven will be in the top 10%.--JohnBlackburne^words_deeds 18:17, 24 October 2010 (UTC)

Nit-picking, but would the fact that you are not replacing the chosen pinapple make any difference? 92.28.246.6 (talk) 20:12, 24 October 2010 (UTC)

Yes, if the qualities are not independently, identically distributed (we may discard this difficulty here, I guess) Pallida  Mors 20:21, 24 October 2010 (UTC)

The questioner stated "top 10% of quality", not "top 10% of the population of pineapples." So, if quality goes from 0 to 99, he wants a quality of 90-99. If the distribution is skewed towards 99, more than 10% of the population will have the top 10% of quality. -- kainaw™ 23:00, 24 October 2010 (UTC)

"top x percent of quality". I normally read that as the corresponding percentile. However, your point can be answered, considering the nth order statistic distribution, computing $1-F^{n}\left(x*\right)$, the probability that the highest of the n pineapples extracted has a quality as good as the threshold x* [again, considering the iid assumption]. (Forgotten sign of Pallida  Mors 00:04, 25 October 2010 (UTC))

A relevant article is order statistic. Pallida  Mors 19:53, 24 October 2010 (UTC)

This is not the same as the secretary problem, or the wives problem suggested by 84.153.., since with the pineapples you can choose any of the pineapples you have previously inspected. Its more like a quality control (poor article) or Sampling (statistics) problem. 92.28.246.6 (talk) 20:03, 24 October 2010 (UTC)

Nitpicking: If the whole pile is N pineapples, and K of them are within the top ten percent of quality of the whole pile, and you buy n pineapples, then the probability that k of these are in the top ten percent of quality is

${\frac {{\binom {K}{k}}{\binom {N-K}{n-k}}}{\binom {N}{n}}}$

See hypergeometric distribution. Bo Jacoby (talk) 22:14, 24 October 2010 (UTC).