Wikipedia:Reference desk/Archives/Mathematics/2008 April 13

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April 13

Not quite the Gamma function...

Hi everyone. Is there an elegant way of dealing with an integral that looks like this?

${\tilde {\Gamma }}(z)=\int _{0}^{\infty }t^{z-1}e^{-it}\,dt$

I'd like to take the real/imaginary parts of this in the end. Any ideas? --HappyCamper 06:22, 13 April 2008 (UTC)

I suggest trying a change of variables, such as $u=it$ (and thus $t=-iu$, $dt=-idu$ and $t^{z-1}=(-iu)^{z-1}=(-i)^{z-1}u^{z-1}$, etc.). --Prestidigitator (talk) 08:42, 13 April 2008 (UTC)

How many hours?

How many hours in 90 minutes? 60 minutes equals 1hr. If you cross multiply then 90 minutes equals 1.5 How do you make 1.5 an hour?71.142.208.226 (talk) 07:13, 13 April 2008 (UTC)Cardinal Raven

It's an hour and a half, that is, one hour plus half an hour. -- Meni Rosenfeld (talk) 07:20, 13 April 2008 (UTC)

Do you mean how to convert a number of hours into the relevant number of hours and minutes? The whole number part (here, the 1) is the number of hours. Multiply the rest by sixty (0.5 * 60 = 30) to give the number of minutes. Daniel (‽) 18:32, 13 April 2008 (UTC)

I believe the question concerns dimensional analysis. Here's why your intuition that the units should work out is indeed correct:

90 min × 1/60 h/min = 90/60 h = 1.5 h.

The general algorithm is to start with the quantity you're given and repeatedly multiply by conversion factors. So long as each new multiplier equals one, you know your final product equals your initial quantity. As another example, consider the problem of expressing the speed 45 ft/s in miles per hour:

45 ft/s × 60 s/min × 60 min/h × 1/5280 mi/ft ≈ 30.7 mi/h.

PaulTanenbaum (talk) 00:33, 17 April 2008 (UTC)

Sum

when (1/e)+(1/f)+(1/g)=0 and e+f+g=3, e^2+f^2+g^2=? the answer is 9, but why? note: ^ means the exponent. —Preceding unsigned comment added by Invisiblebug590 (talk • contribs) 08:03, 13 April 2008 (UTC)

Try multiplying the first equation by efg and squaring the second one. -- Meni Rosenfeld (talk) 08:27, 13 April 2008 (UTC)

Integration Problem

Hi, I'm wondering how to solve this integral: Given that E is the solid bounded by $y=2x^{2}+2z^{2}$ and the plane $y=8$, find $\displaystyle \int \displaystyle \int \displaystyle \int _{E}{\sqrt {3x^{2}+3z^{2}}}\,dV$ Thanks guys! —Preceding unsigned comment added by 70.111.95.226 (talk) 13:31, 13 April 2008 (UTC)

Take the projection of E on the x-z plane. For every point in the projection, the function is constant with respect to y. This will leave you with a 2-dimensional integral, which is best solved with polar coordinates. -- Meni Rosenfeld (talk) 14:14, 13 April 2008 (UTC)

Statistics problems

Hello! I ran into a couple of statistics problems, and wanted to check if my reasoning is correct.

1. Many public schools are implementing a "no pass, no play" rule for athletes. Under this system, a student who fails a course is disqualified from participating in extracurricular activities during the next grading period. Suppose the probability that an athlete who has not previously been disqualified will be disqualified is 0.15 and the probability that an athlete who has been disqualified will be disqualified again in the next time period is 0.5. If 30% of the athletes have been disqualified before, what is the unconditional probability that an athlete will be disqualified during the next grading period?

If we denote the event "an athlete is disqualified" with D, and the event "an athlete had been disqualified previously" with P, then we get that

$P(D|P^{c})=0.15$

$P(D|P)=0.5$

$P(P)=0.3$

$P(D)=?$

And I suppose that $P(D)=P(D|P^{c})\times P(P^{c})+P(D|P)\times P(P)=0.255$

2. A research physician compared the effectiveness of two blood pressure drugs A and B by administering the two drugs to each of four pairs of identical twins. Drug A was given to one member of a pair; drug B to the other. If, in fact, there is no difference in the effects of the drugs, what is the probability that the drop in the blood pressure reading for drug A exceeds the corresponding drop in the reading for drug B for all four pairs of twins? Suppose drug B created a greater drop in blood pressure than drug A for each of the four pairs of twins. Do you think this provides sufficient evidence to indicate that drug B is more effectivfe in lowering blood pressure than drug A?

If I understand correctly, the probability that the reading for drug A exceeds the reading for drug B in a single pair of twins is 1/2. Therefore, the probability that the reading for drug A exceeds the reading for drug B should be $(1/2)^{4}=1/16$. Regarding the second question, I think that this is not enough evidence, as there is a 1/16 chance that this was just a coincidence.

Why I am wondering if my reasoning in these problems is correct is that these solutions seem far too easy, especially my solution to the second problem. So, if you find any errors, please point me towards them :)  ARTYOM  14:28, 13 April 2008 (UTC)

For problem 1, the formula is correct but the number is wrong - I got 0.255.

For problem 2, indeed assuming that the probability of an equal blood pressure drop is 0, symmetry guarantees that the probability is 1/2 each and 1/16 total. I agree that there is not enough evidence, but not for the reason you stated - the chance that this was a coincidence is not 1/16, in fact, we can't calculate it without knowing the prior probability of the treatments being different. -- Meni Rosenfeld (talk) 14:41, 13 April 2008 (UTC)

Thanks a lot! I got 0.255 for the first problem now too; no idea how I came up with 0.0561 the first time :D  ARTYOM  15:01, 13 April 2008 (UTC)

In future, it's good to "sanity check" your answers. What you're doing is taking a weighted average, so the correct answer must be somewhere between the two numbers you're averaging. Your answer was less than both, so couldn't possibly be right. There are often quick checks like that with maths problems - everyone makes mistakes, so it's good to check your answers are sane. --Tango (talk) 21:42, 13 April 2008 (UTC)

Perhaps the answer that the first part of problem 2 is "looking for" is 1/16, but this is incorrect. It can't be calculated using only the information given. Think about it: if the reading for drug "A" exceeds the reading for drug "B" on the first three sets of twins, your expectation that it will as well for the last set of twins should be greater than 1/2.

Only if you know beforehand that drugs "A" and "B" are equally effective (i.e. the probability of either one doing better in a given pair-of-twins trial is 1/2) is the answer 1/16. kfgauss (talk) 09:44, 14 April 2008 (UTC)

It is stated explicitly that we are assuming the treatments are equally effective. The question would obviously be meaningless otherwise. -- Meni Rosenfeld (talk) 10:02, 14 April 2008 (UTC)

Ah, reading the problem. How novel :). In any case, the concern I mention is relevant for the second half of the problem. As you say, having a prior probability distribution is key. If you live in a world in which two drugs are rarely of comparable effectiveness, then 4 trials may be fairly convincing. If you live in a world in which a two drugs are usually comparably effective, then you may need many many trials to determine with confidence which is better. A good reference for this stuff is Example 3 in Bayes'_theorem. kfgauss (talk) 18:50, 14 April 2008 (UTC)

Soon enough, we'll have probabilistically checkable problems, for which only 3 bits of the problem need to be read in order to know the solution :) -- Meni Rosenfeld (talk) 19:20, 14 April 2008 (UTC)

I didn't like the way question 1 was written. Do they mean the chances a particular athlete will be disqualified (that's how everyone here read it), or of any athlete in the class, school, or district being disqualified ? For the second interpretation you'd need to know the number of athletes in the group.

One other assumption for part 2 seems to be that the twins can't have identical BP readings. StuRat (talk) 19:02, 14 April 2008 (UTC)

Regarding the athletes problem - as far as I understand, your second interpretation is what the problem asks. However, despite the fact that I'm not "friends" with statistics (not at all! :D), I don't think that we have to know the total number of athletes in the group, since we are given the information that 30% had been disqualified before. And regarding the second problem--as far as I understand, again--they say that the drop in the BP is measured and compared, not the BP itself. The twins are taken because, I suppose, there is a great probability that the same medicine will have quite similar impact on both.  ARTYOM  21:12, 14 April 2008 (UTC)

No, the chances that at least one athlete would be disqualified would go up the more people you have in the group. As for measuring the drop in BP, twins could still have identical drops, at least to the accuracy measured by the apparatus. StuRat (talk) 02:34, 15 April 2008 (UTC)

And yet another question is giving me a hard time!

3. A survey of people in a given region showed that 20% were smokers. The probability of death due to lung cancer, given that a person smoked, was roughly 10 times the probability of death due to lung cancer, given that a person did not smoke. If the probability of death due to lung cancer in the region is 0.006, what is the probability of death due to lung cancer given that a person is a smoker?

Initially this seemed like an easy problem for me. I denoted "smokers" event with S, and the "death due to lung cancer" with L. Then from the question we get that

$P(S)=0.2$

$P(L|S)=10\times P(L|S^{c})$

$P(L)=0.006$

$P(L|S)=?$

So I went on with

$P(L)=P(L|S)+P(L|S^{c})=11P(L|S^{c})=0.006$

Then, $P(L|S^{c})\approx 0.00055$, so $P(L|S)=0.0055$

But then I figured that perhaps the solution should be similar to the solution of problem #1 above (about athletes), and that $P(L)=0.006$ should in fact equal to $P(L|S)\times P(S)+P(L|S^{c})\times P(S^{c})$, which in the end gives me that $P(L|S)\approx 0.0214$. Which of these ways is the right way, as both of them make perfect sense to me, and why?  ARTYOM  21:32, 14 April 2008 (UTC)

I figure it this way:

P_n = Probability that a given Nonsmoker dies of lung cancer.

P_s = 10P_n = Probability that a given Smoker dies of lung cancer.

P_? = Probability that a given person of unknown smoking status dies of lung cancer.

Since we know 20% of the people are smokers, we get:

P_? = (0.2)P_s + (0.8)P_n

P_? = (0.2)10P_n + (0.8)P_n

P_? = 2P_n + (0.8)P_n

P_? = 2.8P_n

Now we toss in the actual value:

2.8P_n = 0.006

P_n = 0.0021428

This gives us:

P_s = 10P_n = 10(0.0021428) = 0.021428

Or, about a 2 1/7 % chance, which matches your second result. Your first result can be dismissed because you calculated the chances of a smoker dying of lung cancer at being less than those of the average person, and that's obviously wrong. You didn't even use the 20% smokers figure in the calculations, and that's obviously important. StuRat (talk) 03:09, 15 April 2008 (UTC)

Unknown Symbol

The other day I was at a science fair and saw a formula pertaining to gravity and general relativity. I can't remember the exact formula but it had the general format of ${\frac {a-b}{c^{n}+d}}{\Bigg |}_{x}^{y}$. What is the meaning of the | ? Thanks, Zrs 12 (talk) 18:16, 13 April 2008 (UTC)

It means "evaluated at", or in the case of a number at the top and bottom, "evaluated between". You evaluate it at the top value, and then subtract it's value at the bottom value. In the case of physics, the bottom value was probably the starting point, and the top value the end point. The energy required to lift something, for example, is the gravitational potential energy at the end minus the gravitational potential energy at the beginning - that could be written using such notation. --Tango (talk) 19:29, 13 April 2008 (UTC)

So $\int _{1}^{5}2x\ \operatorname {d} x=x^{2}|_{1}^{5}=F(5)-F(1)=24$? Zrs 12 (talk) 19:48, 13 April 2008 (UTC)

Yep. --Tango (talk) 19:59, 13 April 2008 (UTC)

Thanks, Tango Zrs 12 (talk) 23:02, 13 April 2008 (UTC)

Only that it should be $\int _{1}^{5}2x\ dx$. The dx is an integral part (pun intended) of the notation. -- Meni Rosenfeld (talk) 23:17, 13 April 2008 (UTC)

Haha. Nice one. Yep, I'm not in calculus yet so I tend to forget the dx quite often. Zrs 12 (talk) 00:32, 14 April 2008 (UTC)

Square Roots

Can anyone give me a way to find square roots by hand without the guess and check method? Zrs 12 (talk) 19:00, 13 April 2008 (UTC)

Take a look at Methods of computing square roots, in particular Methods of computing square roots#Digit by digit calculation. -- Meni Rosenfeld (talk) 19:30, 13 April 2008 (UTC)

Thanks, Meni Zrs 12 (talk) 23:04, 13 April 2008 (UTC)

Finding digits of Pi

Is it possible to calculate any digit of Pi (or other types of mathematical constants) without knowing the previous digits in the decimal expansion ? How/Why not ? -- Xedi (talk) 21:19, 13 April 2008 (UTC)

See Bailey-Borwein-Plouffe formula. PrimeHunter (talk) 21:29, 13 April 2008 (UTC)

Yes, for some constants and in some bases - this is called a spigot algorithm. As PrimeHunter says, the Bailey-Borwein-Plouffe formula is a spigot algorithm for the binary expansion of π. I don't know whether a spigot algorithm is known for the decimal expansion of π. Gandalf61 (talk) 21:36, 13 April 2008 (UTC)

See also, Science News Online, Ivars Peterson's MathTrek (2/28/98): Pick a Digit, Any Digit. – b_jonas 08:53, 14 April 2008 (UTC)

Thanks. -- Xedi (talk) 15:50, 14 April 2008 (UTC)