Wikipedia:Reference desk/Archives/Mathematics/2011 September 9

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September 9

What does the Standard Deviation represent in laymans terms

Say you have two gambling games. Game A has an in-built house edge of 1% and a SD of 1. Game B has an in-built house edge 2% and an SD of 2. So we can say that over the long term, a player will lose twice as much playing Game B as they will playing Game A, since the house edge is exactly double. So far pretty simple... But what I cannot get my head around, is what the SD figure ACTUALLY represents. In school and college we did a fair bit of Standard Deviation, and I always knew how to calculate it, but never how to interpret it. So from the example above, based on the data I have given, what can we deduce from comparing the SD figures of Game A and B?

All I am able to deduce from it is that Game B is more volatile, and has a wider spread of likely outcomes, which means the players winnings, or more likely losses will likely be greater/less than on Game A. However I guess it would be wrong to assert that Game B as double the possible range of outcomes, since anything could potentially happen.

I would really appreciate it if someone could help to clarify what SD figures actually denote! Flaming Ferrari (talk) 01:50, 9 September 2011 (UTC)

It means that, given one set of people for each game that have all played the same number of games, the average (absolute) difference between a given player's take and the average take for game B will be twice that average difference for game A. The possible range (in the long run) is not different, as you say, but the distribution of results per-player is twice as wide. (That distribution also widens with the number of games played; that's why I said "played the same number of games".) --Tardis (talk) 07:17, 9 September 2011 (UTC)

Agreed, except that mean absolute deviation is not identical to standard deviation Dbfirs 11:32, 11 September 2011 (UTC)

Another way to view it is by looking at a normal distribution curve - which helps those that like to visually identify concepts. See Image:Normal Distribution PDF.svg. The average for the red and blue curves is the same. However, the standard distribution is different. Red has a wider standard distribution. So, if you consider the population of events (player's games in your case) to be represented by the curve, you can imagine that each player's game is bounded under the curve (and above the X=0 line). So, in the blue curve, it is very likely that they will be near the average since most of the room under the curve is near the average. In the red curve, there is more probability that it can fall further from the average. -- kainaw™ 13:06, 9 September 2011 (UTC)

Put as briefly as possible, it represents the average deviation from the mean -- except that in order to deal with the fact that deviations can be either positive or negative, the average is taken as a root mean square. Looie496 (talk) 14:12, 9 September 2011 (UTC)

Another approximate way of looking at SD (for a normal distribution) is to say that about two thirds of the results will be within one standard deviation of the mean, and more than 95% of results will be within two SDs of the mean. Dbfirs 17:04, 10 September 2011 (UTC)

Hypergeometric Series

Could some helpful person, most likely owning a copy of Mathematica, possibly evaluate the following as I keep getting bogged down in hypergeometric series. I think the hypergeometrics gained from evaluating one of the sums become jacobi polynomials which compined with the second summation becomes a generating function, and finally the last summation is then simply a geometric series. However I keep slipping and making errors.

$\sum _{i=1}^{\infty }\sum _{j=1}^{\infty }\sum _{n=\mathrm {max} (i,j)}^{\infty }a^{i}b^{j}c^{n}{\binom {n}{i}}{\binom {n}{j}}$

Where of course {a,b,c} are reals between 0 and 1. Thank you. — Preceding unsigned comment added by 192.76.7.237 (talk) 13:21, 9 September 2011 (UTC)

I don't own Mathematica but with pen and paper I get

${\frac {1}{1-(a+1)(b+1)c}}-{\frac {1}{1-(a+1)c}}-{\frac {1}{1-(b+1)c}}+{\frac {1}{1-c}}$

--RDBury (talk) 14:51, 9 September 2011 (UTC)

For what it's worth, the Mathematica input

Sum[
  Sum[
    Sum[
        a^i*b^j*c^n*Binomial[n, i]*Binomial[n, j],
    {n, Max[i, j], Infinity}],
  {j, 1, Infinity}],
{i, 1, Infinity}]

gives me the output

Sum[
  Sum[
      a^i*b^j*c^Max[i, j]*Binomial[Max[i, j], i]*Binomial[Max[i, j], j]*
      HypergeometricPFQ[{1, 1 + Max[i, j], 1 + Max[i, j]}, {1 - i + Max[i, j], 1 - j + Max[i, j]}, c],
  {j, 1, Infinity}], 
{i, 1, Infinity}]

In other words, it only seems to manage the innermost sum. 130.88.73.71 (talk) 14:57, 9 September 2011 (UTC)

How about going the other way? I.e. expand the expression I gave into a power series to see if it matches the original sum.--RDBury (talk) 16:31, 9 September 2011 (UTC)

Alternatively, revert the series (I don't recall the TeX align symbols, so I'm just going to produce new equal expressions on each line):

$\sum _{n=1}^{\infty }\sum _{i=1}^{n}\sum _{j=1}^{n}a^{i}b^{j}c^{n}{\binom {n}{i}}{\binom {n}{j}}$

$\sum _{n=0}^{\infty }\sum _{i=1}^{n}\sum _{j=1}^{n}a^{i}b^{j}c^{n}{\binom {n}{i}}{\binom {n}{j}}$

$\sum _{n=0}^{\infty }\left(\sum _{i=1}^{n}{\binom {n}{i}}a^{i}\right)\left(\sum _{j=1}^{n}{\binom {n}{j}}b^{j}\right)c^{n}$

$\sum _{n=0}^{\infty }\left((a+1)^{n}-1\right)\left((b+1)^{n}-1\right)c^{n}$

${\frac {1}{1-(a+1)(b+1)c}}-{\frac {1}{1-(a+1)c}}-{\frac {1}{1-(b+1)c}}+{\frac {1}{1-c}}$

— Arthur Rubin (talk) 16:26, 10 September 2011 (UTC)

I have taken the liberty of fixing a couple of small errors in one of your expressions -- hope you don't object. Looie496 (talk) 16:58, 10 September 2011 (UTC)

Let me also note that it isn't entirely clear that the reversion is valid -- the reverted sum does not converge for all values with 0<a<1 and 0<b<1. Looie496 (talk) 17:04, 10 September 2011 (UTC)

True. The reverted sum converges (if a, b, and c are positive) only if (a+1)(b+1)c < 1. On the other hand, if a, b, and c are positive, then all terms are positive, so the reversion is valid. — Arthur Rubin (talk) 04:04, 11 September 2011 (UTC)

Inverse of 4th degree polynomial

Does a general fourth-degree polynomial have a closed-form inverse? I know there are closed forms for the roots of polynomials up to the fourth degree (but not fifth). Maple will give a closed form inverse for second and third degree polynomials but not for a fourth degree polynomial. Does a closed form for it exist? Bubba73 ^{You talkin' to me?} 15:14, 9 September 2011 (UTC)

I believe it would be an application of Ferrari's solution of Quartic equations, so technically the answer is yes. But the expression would be so complicated that I'm not surprised Maple wouldn't give it to you.--RDBury (talk) 16:20, 9 September 2011 (UTC)

Thanks.

Resolved

Bubba73 ^{You talkin' to me?} 01:36, 10 September 2011 (UTC)

mathematics

I am doing on a paper. I have used several pieces of information found in Wikipedia/Mathematics & Wikipedia/Science. If I want to copywrite this work ,should I reference Wikipedia? Thank you. — Preceding unsigned comment added by Arthur204 (talk • contribs) 21:15, 9 September 2011 (UTC)

I am not a lawyer and am not supposed to give legal advice per se. However the general principle (at least in the US) is that facts are not copyrightable. So if you're just using the facts, as opposed to the words (or drawings, images, etc), there should be no copyright difficulty.

A completely separate question from the legal one of copyright is the ethical one of giving credit. For any information you get from Wikipedia or any other source, you have a responsibility not to make it appear as though it's your own work. --Trovatore (talk) 21:20, 9 September 2011 (UTC)

It says at the top of the page that this isn't the place to get opinions or advice. This is more a WP policy question than a math question.--RDBury (talk) 23:26, 9 September 2011 (UTC)

If your paper consists entirely of material from Wikipedia, then it cannot be copyrighted, but must be released under free licence (see Wikipedia:Copyrights). It is much more likely that your paper contains original work, and just uses some material from Wikipedia. In this case your original work is copyright, but you should acknowledge all use of material provided by others, as explained above. Dbfirs 17:43, 10 September 2011 (UTC)