Wikipedia:Reference desk/Archives/Mathematics/2009 August 24

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August 24

Probability

I've got a question on Probability which I found on one of my reference books, which I couldn't figure out. Guys, this is not homework, just an interesting question which I ran into. Say a police officer is investigating a case. He is 60% convinced that X is the culprit. Suddenly, he finds a new piece of evidence, which is a mark on the culprit's face, which 90% of the population don't have. If X has the mark, what is the probability that X is the prisoner ? Actually, though this question looks simple enough, I just can't crack it ! Excuse me if I had asked a very obvious question, but I need some help. Rkr1991 ^{(Wanna chat?)} 10:54, 24 August 2009 (UTC)

Let A be the event "X is the culprit" and B "X has a mark". We have $P(A)=0.6,P(B|A)=1,P(B|A^{C})=0.1\,\!$. By Bayes' theorem the posterior probability of A given that we know B is

$P(A|B)={\frac {P(A)P(B|A)}{P(B)}}={\frac {0.6\cdot 1}{P(A,B)+P(A^{C},B)}}={\frac {0.6}{0.6+0.04}}=0.9375$

-- Meni Rosenfeld (talk) 12:16, 24 August 2009 (UTC)

Either there's a typo in the question or it's a trick question. The probability that X is the prisoner cannot be calculated from the police officer's certainty that X is the culprit and the ratio of people carrying a certain facial mark. We're not told anything about any trial turning a suspected culprit into a prisoner. ~~ Dr Dec (Talk) ~~ 12:24, 24 August 2009 (UTC)

Great, that's the right answer. Thanks a lot !!!! Rkr1991 ^{(Wanna chat?)} 13:25, 24 August 2009 (UTC)

General formula for power sum

Today I am the one who asks about a general formula for the following sum, of integers.

$\sum _{i=1}^{n}i^{k}$

For example, the simple form is given by the arithmetic progression sum, in which k = 1, is given as

$\sum _{i=1}^{n}i={\frac {n+n^{2}}{2}}$

If there exists an article, I'd be glad to be guided to.--Email4mobile (talk) 16:27, 24 August 2009 (UTC)

Have a look at Bernoulli number which answers your question. 129.67.186.200 (talk) 16:39, 24 August 2009 (UTC)

Thank you very much, 129.67.186.200. That is what I need. I did try to find a general solution yesterday, didn't know this formula exists. That's my try, User:Email4mobile/sum of power arithmetic progression.--Email4mobile (talk) 21:46, 24 August 2009 (UTC)

Also: Faulhaber's formula. Michael Hardy (talk) 23:15, 24 August 2009 (UTC)

Conceptually the simplest way to figure out that summation is through the Lagrange interpolation formula. You know that it's obviously a polynomial of degree k+1, so plot some points and then fit the curve through it. 70.90.174.101 (talk) 00:14, 25 August 2009 (UTC)

Greater, lesser or non-comparable

Would it be proper to say that negative numbers are less than positive numbers -- because an angle of -7 is a greater angle than an angle of 5? Or are angles different, because in regard to angles magnitude is all that matters? DRosenbach ^{(Talk | Contribs)} 18:55, 24 August 2009 (UTC)

Yes: the negative numbers are less than the positive numbers, with repect the the usual ordering. What exactly do you mean when you say that "−7 is a greater angle than an angle of 5"? I assume you mean something along the lines of an angle of −7 degrees is larger than an angle of 5 degrees. Well, this is very different. The number −7 is further from 0 than 5 is. This is because |−7| > |5|. It all depends on how you chose to order your set, and once an ordering has been chosen then it doesn't really make any sense to compare orderings. For example, given a natural number k, the metric norm

$||(x,y)||_{k}:=(x^{k}+y^{k})^{1/k}\,$

have very different "unit disks". The unit disk for k = 2 (i.e. in the usual Euclidean metric) is the usual circular Euclidean unit disk. In the case where $k\to \infty$, the unit ball is the Euclidean unit square! ~~ Dr Dec (Talk) ~~ 19:27, 24 August 2009 (UTC)

Wow...I got everything up until your example of "k" -- from then on, I would say it's all Chinese to me. If you insist it's English, then I don't know what I'll say... :) DRosenbach ^{(Talk | Contribs)} 21:59, 24 August 2009 (UTC)

I think it was just an analogy. The real numbers with different orders are completely different things in the same way that the plane with different metrics/norms are completely different things. It doesn't really matter, but basically that formula is a new definition of the distance between two points, so you end up with up different definitions of discs (since a disc is all the points less than or equal to a certain distance from a certain point, so you chance the definition of distance you change the definition of a disc). If it doesn't make sense, just ignore it! The important bit is everything before "For example". --Tango (talk) 22:14, 24 August 2009 (UTC)

Well, there are many ways to measure distance, and to measure the length of something. A metric is something that we use to measure the distance between two points. Let's consider the plane, denoted by $\mathbb {R} ^{2}$, as an example. How do we define the distance between two points, say a and b? Well, if a = (a₁,a₂) and b = (b₁,b₂) then the distance, let us denote it by d(a,b), in the good old fashioned using-a-ruler sense is given by

$d({\mathbf {a} },{\mathbf {b} })={\sqrt {(a_{1}-b_{1})^{2}+(a_{2}-b_{2})^{2}}}\ .$

The idea of distance has some nice properties. Let a, b and c be points in the plane (not nevessarily distinct).

1. The distance from a point to itself is zero, i.e. d(a,a) = 0.
2. The distance between points is always non-negative, i.e. d(a,b) ≥ 0.
3. Informally, the shortest distance between two points is a straight line, i.e. d(a,b) + d(b,c) ≥ d(a,c).

This last one is called the triangle inequality. Well, what we do is to abstract this idea. We say that any function d that satisfies these three conditions is a metric, and gives us some idea of distance. One such metric would be

$d_{4}({\mathbf {a} },{\mathbf {b} })=((a_{1}-b_{1})^{4}+(a_{2}-b_{2})^{4})^{1/4}\ .$

In fact, for any positive integer k we have have a metric given by

$d_{k}({\mathbf {a} },{\mathbf {b} })=(|a_{1}-b_{1}|^{k}+|a_{2}-b_{2}|^{k})^{1/k}\ .$

Good old fashioned distance comes from k = 2. But for all positive integers k we have a metric, i.e. a function that takes any two points in the plane and gives us a number in a way that satisfies our three conditions above. Now, the unit disk, for a metric d, centred at some point a is given by all the points b judged to be at a distance of less than or equal to 1 from a, i.e.

$D_{\mathbf {a} }=\{{\mathbf {b} }\in \mathbb {R} ^{2}:d({\mathbf {a} },{\mathbf {b} })\leq 1\}\ .$

In the case of d₂ the unit disk, centred at the origin is just the set of all points at a distance at most 1 from the origin, i.e. the filled in circle. This matches our idea of equidistance: the points at a fixed distance from you will form a circle - a circle has a constant radius. But the unit disks for the metrics d_k are far from circular disks. In fact, as k gets bigger and bigger, the disks get more and more square shaped. In fact the limiting shape is exactly a square. So, according to the metric d_∞, the points a fixed distance from the origin actually form a square! The point is that there are many ways of talking about size and distance. Some are very natural, like d₂; but some are most counterintuative, like d_∞. You need to use the right metric for the right problem, and be very careful when you're trying to compare answers in one metric to answers in another metric, or else you might start saying that a square is a circle (and it is, topologically speaking... but that's another story). ~~ Dr Dec (Talk) ~~ 22:49, 24 August 2009 (UTC)

With angles it depends on what you are doing. 350 degrees and -10 degrees are often completely equivalent, but have very different magnitudes. Sometimes you want to consider the magnitudes, sometimes you want to consider them modulo 360 degrees, sometimes you want the actually real number itself. --Tango (talk) 20:07, 24 August 2009 (UTC)

To some extent, the original question appears to be a language-usage question rather than a mathematical one. Do we say −1000000 is a bigger or smaller negative number than −1? I'm afraid authors are probably inconsistent on this point. But if you see the phrase large negative number it most likely means "negative number of large absolute value" rather than "negative number far to the right (among negative numbers) in the number line". --Trovatore (talk) 22:54, 24 August 2009 (UTC)