Wikipedia:Reference desk/Archives/Mathematics/2012 September 27

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September 27

mode

when calculating a mode, what do you do when 1: there is no number that occurs more then once, and 2: there is a tie between 2 or more numbers (example: 1, 2, 3, 3, 4, 4, 5)? p.s.: are your captchas cap-sensitive? thanks, 70.114.254.43 (talk) 00:51, 27 September 2012 (UTC)

As for the mode, those are probably situations where the mode is not an appropriate measure of central tendency. It's meant for data where there is a clear mode, like say a (possibly distorted) bell-shaped curve. Therefore, the mode should be either a single value, or perhaps two adjacent values, in which case you could say the mode is midway between the two, so 3.5, in your example. StuRat (talk) 00:57, 27 September 2012 (UTC)

I don't know for certain if the capcha used when you sign up for a Wikipedia account is case-sensitive, but it would make sense if it were, to make it more difficult for programs to determine. StuRat (talk) 01:00, 27 September 2012 (UTC)

The term bimodal is frequently used if there are two most-prevalent values. Extensions to trimodal, etc. are obvious, but much less frequently used (usually lumped into the general category of "multimodal") - note that a set/distribution with a single mode would be classed as unimodal. Note that there wouldn't be a single value for the mode in that case. For a set/distribution without a mode, the use of Latin prefixes would mean the distribution would probably classed as "nonmodal" (as opposed to the greek-derived "amodal"), and if you were looking for a value of the mode, it would be undefined. - Note that having an undefined central tendency value isn't a problem. There are continuous distributions (like the Cauchy distribution) where the mean is not defined. -- 205.175.124.30 (talk) 01:59, 27 September 2012 (UTC)

• Modes are mainly useful for characterizing mathematical distributions, such as a Gaussian etc. They are essentially useless for characterizing data unless you first fit a mathematical distribution to the data. Calculating modes directly from empirical data is almost always a waste of effort. Looie496 (talk) 02:34, 27 September 2012 (UTC)

A median can be mid-way between two values, but NOT a mode (unless Stu was thinking of making a rough estimate of the mode of a continuous unimodal distribution from a sample of values). There can be two different modes (bimodal, as mentioned above). Dbfirs 10:05, 27 September 2012 (UTC)

Yes, I'm thinking of an example where you plot incomes for a population of people and find the two most common incomes are $26,395 and $26,396. In a case like this, I think you would be justified in just calling the mode $26,395.50. StuRat (talk) 02:56, 28 September 2012 (UTC)

To enlarge slightly on my earlier point, the main place where modes come into play is in constructing histograms from data -- the mode is reported as the highest bin in the histogram. I repeat that calculating modes from raw data is almost always useless, unless the raw data is seriously discretized. Looie496 (talk) 03:03, 28 September 2012 (UTC)

Agreed on both points. The example given by the OP was presumably an artificial one at a very elementary level. Dbfirs 06:46, 28 September 2012 (UTC)

Uniformly bounded

Show that $\left|\int _{0}^{xN}{\frac {\sin u}{u}}du\right|+|x|$ is uniformly bounded in $x\in [-\pi ,\pi ]$ and $N\in \mathbb {N}$, using the fact that $\int _{0}^{\infty }{\frac {\sin u}{u}}du$ converges.
Boundedness is clear to me; it is only uniform boundedess which is unclear to me. Widener (talk) 19:20, 27 September 2012 (UTC)

I think it means $\exists M:\forall N\in \mathbb {N} ,\forall x\in [-\pi ,\pi ]:\left|\int _{0}^{xN}{\frac {\sin u}{u}}du\right|+|x|<M$

I'd try something like:

$\int _{0}^{\infty }{\frac {\sin u}{u}}du=\lim _{s\to 0}\int _{s}^{1}{\frac {\sin u}{u}}du+\lim _{t\to \infty }\int _{1}^{t}{\frac {\sin u}{u}}du=\lim _{s\to 0}f(s)+\lim _{t\to \infty }g(t)=F+G$

Since the integral converges, the two limits will as well.

Let's find a bound for g(t):

sin/x is continuous over [1, inf] and so is g(t). g(t) converges to G; then from definition of limit: $\exists D:\forall t>D:\left|g(t)\right|<\left|G\right|+1$; and since the function is continuous in [1,D] you can use the extreme value theorem, $\exists E>0:\forall t\in [1,D]:\left|g(t)\right|<\left|E\right|$ take K=max(|G|+1,E) and you've got: $\forall t\in [1,\infty ]:|g(t)|<K$

Do the same for f(s), gives you a bound L;

M=K+L+pi , a bound for the whole function with Nx in [0,infinity]

since sin(-x)/(-x)=sin(x)/x the same values are valid for [-inf,0], so for all possible values xN... Ssscienccce (talk) 13:32, 29 September 2012 (UTC)

Sigmoid function between two points

Hi, I'm wondering how to produce a sigmoid function that "starts" and "ends" at two points in a graph. So, for example, if I have p1=(2,2) and p2=(10,3), I would like to create a sigmoid function such that we pass through both these points, and the gradient is (or is nearly) 0 at each point, and the gradient is maximal exactly half-way between the points. Thanks! — Sam 63.138.152.135 (talk) 19:27, 27 September 2012 (UTC)

See Sigmoid function. Choose your base function (eg y = 1 / 1 + e^-x) and scale and offset it to go through your points. That particular function is a problem in that it doesn't reach the unscaled limits of 0 and 1 until -/+ infinity. At -/+ infinity you get a slope of 0, but it can't be scaled. You could choose a point close to the limit and say the tolerance is good enough or you could determine some finite exact points and scale those for a slope of only almost zero. Alternatively you choose a different function - there are many that have that sort of shape, but whether any have slope 0 at finite x I don't know. -- SGBailey (talk) 21:49, 27 September 2012 (UTC)

If you need to do it exactly, use spline interpolation with a cubic spline -- see Spline interpolation#Algorithm to find the interpolating cubic spline for the formula. Looie496 (talk) 22:18, 27 September 2012 (UTC)