Wikipedia:Reference desk/Archives/Mathematics/2010 June 28

= June 28 =

Bayesian inference with very little information
See the subheading "False positives in a medical test" in the article Bayesian inference. How would the problem be solved (i.e. the probability of a false positive) if P(B|A), P(A), P(B) were completely unknown?––115.178.29.142 (talk) 00:52, 28 June 2010 (UTC)
 * The result $$\scriptstyle \frac {\frac 1 2 \frac 1 2}{\frac 1 2}=\frac 1 2$$ is as unreasonable as the assumption. Bo Jacoby (talk) 04:27, 28 June 2010 (UTC).

Exponents and Fractions
I do not understand how to approach this question: $$\frac{(x^{a-1})^3}{(x^{2a})(x)}.$$ The answer is $$x^{a-4}.$$ Thank you for the help. —Preceding unsigned comment added by 24.86.187.199 (talk) 05:25, 28 June 2010 (UTC)


 * To simplify this expression, start by getting both the numerator and the denominator as expressions with a single x then see if it becomes obvious. -- SGBailey (talk) 05:57, 28 June 2010 (UTC)


 * Wikipedia doesn't do your homework. 194.100.253.98 (talk) 05:55, 28 June 2010 (UTC)
 * OP: note that yours "x^a-1" is very ambiguous! it should be possibly read "$$x^a-1$$" but in fact from the answer you wrote, I understand you mean "x^(a-1)", that is $$x^{a-1}$$. I've put the formulas in more readable style. Please try to learn how to write maths formulas, that would make easier for you to get answers. In these cases, you can edit similar formulas and see how they did it. On a different note, please don't forget to sign your post. --pm a  08:57, 28 June 2010 (UTC)


 * See Exponentiation. -- 58.147.53.253 (talk) 12:46, 28 June 2010 (UTC)

Statistical sample
Hello. I have a question concerning statistical samples. I have written a small program that flips two coins each round for 50000 rounds. In order to demonstrate the Boy or Girl paradox, I made it ignore results where both coins turn up to be tails and instead flip the coins again during the same round, so I end up with 50000 legal rolls. Now, I'm wondering about the official sample size of the program - is it 50000 (the amount of legal rolls) or are the ignored rolls also officially part of the sample? 194.100.253.98 (talk) 05:54, 28 June 2010 (UTC)
 * Isn't this the question at the heart of the paradox? Your choice will determine your answer.    D b f i r s   12:51, 28 June 2010 (UTC)
 * Well, I know that the answer is 1/3. It's just that another person claims that while my answer is correct, officially the double tails sets are also a part of the sample (even though ignored as they don't qualify by the problem's criteria). I chose not to include them in the sample size count, since they are "impossible" by the way the problem is defined. Does the sample by definition include discarded, "impossible" random rolls? 194.100.253.98 (talk) 13:01, 28 June 2010 (UTC)
 * You're trying to estimate a probability, or equivalently, the proportion of "rolls with at least one head" where "both rolls are heads". So yes your total sample space is only those rolls that were valid - the two-tailed rolls are not counted. Confusing Manifestation (Say hi!) 00:02, 29 June 2010 (UTC)
 * Thank you! 194.100.253.98 (talk) 05:25, 29 June 2010 (UTC)

General Cardioid Equation
Hey mathletes-

I'm trying to understand cardioid functions in polar coordinate form.

Is the most general form P(theta)=2a(1-cos(theta))? that is to say, all cardioids are of this form?

thanks209.6.54.248 (talk) 15:28, 28 June 2010 (UTC)


 * See Cardioid, yes they are all of that form though of course the equation can change if the shape is moved around. The term is also sometimes applied to any shape that looks like that. Dmcq (talk) 15:58, 28 June 2010 (UTC)

Limaçon
Limaçon translates to mean snail, but why are the Limaçon curves named after snails? •• Fly by Night (talk) 19:27, 28 June 2010 (UTC)
 * They look like snail shells. The resemblance may be a bit strained but that's typical in the naming of curves.--RDBury (talk) 21:37, 28 June 2010 (UTC)
 * I have to disagree. They look nothing like snail shells. Could you please provide an example an image of a snail shell that looks anything like a Limaçon? •• Fly by Night (talk) 21:02, 30 June 2010 (UTC)
 * As the article states, "Depending on the position of the point generating the curve, it may have inner and outer loops (giving the family its name), it may be heart-shaped, or it may be oval." [[Image:Limacons.svg|thumb|500px|none|Three limaçons: the one on the right resembling a snail's shell and its reflection.]] A couple of snail images for comparison are here, here, and here.- ToET 13:43, 4 July 2010 (UTC)

Are these formulas known in the litterature of statistics?
In the above thread the discrete probability distribution $$\scriptstyle P(H|h,N,n) =\frac{\binom H h \binom{N-H}{n-h}}{\binom {N+1}{n+1}}$$ is defined from Bayes' theorem based on the hypergeometric distribution $$\scriptstyle P(h|H,N,n) =\frac{\binom H h \binom{N-H}{n-h}}{\binom {N}{n}}$$ and the marginal distributions $$\scriptstyle P(h|N,n) =\frac{1}$$ and $$\scriptstyle P(H|N,n) =\frac{1}$$. As this computation is very elementary I expected to find the probability distribution $$\scriptstyle P(H|h,N,n)$$ in the litterature, but so far I have searched in vain. So my question is: is it known, or is it original research on my part? If it is known, then please provide a reference and the name of the distribution $$\scriptstyle P(H|h,N,n)$$. (I call $$\scriptstyle P(H|h,N,n)$$ the discrete induction distribution, and the hypergeometric distribution $$\scriptstyle P(h|H,N,n)$$ is the corresponding discrete deduction distribution).

The mean ± standard deviation of the hypergeometric distribution is
 * h ≈ f (H, N, n)

where f is defined by f (H, N, n) = $$\scriptstyle \frac{Hn}N\pm \sqrt{\frac{\frac{Hn}N(1-\frac HN)(1-\frac nN)}{1-\frac 1N}}$$.

The mean ± standard deviation of the induction distribution is
 * H ≈ &minus;1&minus; f (&minus;1&minus;h, &minus;2&minus;n, &minus;2&minus;N).

My proof of this result is boring, but the result itself is fascinating, because it says that the transformation
 * (H, h, N, n) → (&minus;1&minus;h, &minus;1&minus;H, &minus;2&minus;n, &minus;2&minus;N)

translates between induction and deduction. Is this formula known?

In the J (programming language) the deduction and induction formulas are deduc=.(*(%+/))([,:%:@*)(*/@}.%{.)@(-.@((1,,:)%+/@])) induc=.*&_1 1@(+&1 0)@(-@>:@[deduc~-@(+#)~) 1 2 induc 4 1.4     2.6 0.489898 0.489898 The example shows that if you got 1 tail and 2 heads in the first 3 flips, then you should expect 1.4±0.5 tails and 2.6±0.5 heads in a total of 4 flips. Bo Jacoby (talk) 23:18, 28 June 2010 (UTC).