Wikipedia:Reference desk/Archives/Mathematics/2007 September 22

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September 22

Need Major Math Skills Advice

Ok so here is the deal. I am a highschool sophomore and am only at a beginning Algebra 1 level. So basically as of now math is not my best subject. Now in 9 months I want to take a college placement test so that I can just go to college full time in my Junior and Senior year of highschool but getting both high school and college credits. Now the college placement test I will be taking consists of mainly high school Algebra, and Geometry. So to get to the point.. My question is do you guys have any strong advice to getting to the math level I need to be at in 9 months for that test? I basically need to know Algebra and Geometry for it and I need some major advice in what to do. How can I get ahead and learn the math very quickly? Thank you for your time. 06:07, 22 September 2007 (UTC)

I'd recommend getting one or more maths text books at the level you are at, and working through the proofs/problems until you can do the maths easily. Somebody else might be able to recommend books that you can get in your country - practise makes perfect. And you could ask your teacher to recommend some books.87.102.89.127 16:02, 22 September 2007 (UTC)

Although most books I've seen include lots of silly stuff that the teachers skip. You might ask to see a syllabus for a geometry class from your current math teacher, with the page numbers assigned for each day. This will help you to avoid wasting time on the unnecessary parts. Geometry isn't all that dependent on algebra, so you can probably learn both at the same time. StuRat 03:10, 23 September 2007 (UTC)

Is there any way to find out what kind of questions will be on the test you're studying for? Black Carrot 23:52, 24 September 2007 (UTC)

Poisson Distribution

I have two questions about the Poisson Distribution. The probability function of the Poisson Distribution is:

$f(k;\lambda )={\frac {\lambda ^{k}e^{-\lambda }}{k!}},\,\!$

where:

k = the number of events in a given trial

lambda = the mean number of events over all trials

e = e

Question 1). Does lambda follow a Poisson Distribution?

In other words, suppose I conduct an experiment in which I have 1000 trials. In each trial I record the number of events, then I obtain the mean number of events over all 1000 trials.

Then I repeat the experiment 100 times, so I have 100 estimated means. Do those 100 estimated means follow a Poisson Distribution?

Question 2). Is there a formal test for whether a variable follows a Poisson Distribution? I know there are statistical tests that can be used to estimate, for example, whether errors in regression are normally distributed. I am hoping to find a similar test to estimate whether a variable follows a Poisson Distribution. I suppose such a test would be estimating whether the mean of the variable equalled the variance of the variable.

Thanks for any help with this.

Mark W. Miller 18:26, 22 September 2007 (UTC)

I can only answer about question 1; First, don't confuse $\lambda$ which is a "true" (possibly unknown) parameter of a variable's distribution, with estimates of it based on empirical data; Second, the answer is clearly no, since a variable distributed Poissonly can only take integer values, while $\lambda$ and its estimates can take any nonnegative real value. -- Meni Rosenfeld (talk) 18:33, 22 September 2007 (UTC)

The parameter λ is equal to the expected value of the number of events. The quantity 1000m, where m is the observed mean over 1000 trials, is an integer and does follow a Poisson distribution, with parameter 1000λ. You can use Pearson's chi-square test to test the goodness-of-fit between the theoretical and the observed distributions. Make sure you group adjacent cells with low expected frequency and make sure there are enough cells with expected frequencies of 5 or more. For example, if λ = 4.78 and N = 65, you should lump 0 and 1 together, as well as 9 and higher, giving cells {0..1}, {2}, {3}, {4}, {5}, {6}, {7}, {8}, {9..}. Don't forget to subtract an extra 1 from the parameter for degrees of freedom, since λ was estimated from the observations.  --Lambiam 20:35, 22 September 2007 (UTC)

Knowing λ you cannot deduce k exactly, but k assumes nonnegative integer values following a poisson distribution with mean value λ and standard deviation equal to the square root of λ. Knowing k you cannot induce λ exactly, but λ assumes nonnegative real values following a gamma distribution with mean value k+1 and standard deviation equal to the square root of (k+1). Bo Jacoby 21:30, 22 September 2007 (UTC).

Thank you for the replies.

Regarding Question #2 I've looked into the Pearson's chi-square test for estimating goodness-of-fit and I understand that now.

I'm still a little confused about Question #1. I am using emperical data to estimate λ (and would like to estimate its standard error). I know the ks, the number of events per trial or the counts for each of the 1000 trials (or 1000 samples) in an experiment. Therefore, if I understand correctly, λ follows a gamma distribution (and the observed mean count over all trials in the experiment multiplied by the number of trials in the experiment follows a Poisson Distribution).

What confuses me a little bit is that λ would have a mean value of k+1 with a standard deviation of the square root of (k+1). That kind of suggests to me that λ could have several different means within an experiment, since the range of observed counts (the ks) within the 1000 trials could encompass several different values. The counts within a trial might range from 0 to 10. It kind of sounds like you are saying λ can be estimated separately for each trial within an experiment and that the mean λ would range from 1 to 11 if the counts range from 0 to 10.

Sorry about my confusion, and thanks again for the replies. They have been very helpful.

Mark W. Miller 14:46, 23 September 2007 (UTC)

I too find the statement about a distribution followed by λ confusing. That would be a posterior distribution, based on conditional probability, but don't you need a prior distribution for that?  --Lambiam 15:15, 23 September 2007 (UTC)

Answering Mark W. Miller: If you observe totally k events in 1000 trials, then the intensity per 1000 trials is gamma distributed k+1 ± (k+1)^1/2 (using the convenient but slightly nonstandard notational convention that λ≈μ±σ indicates that the mean value of λ is μ and the standard deviation of λ is σ), The intensity per trial is λ≈(k+1±(k+1)^1/2)/1000=(k+1)/1000 ±(k+1)^1/2/1000. So this is the estimate for λ. If the total count in 1000 trials is, say, 5000, then λ≈5.001±0.0707 counts per trial.

Answering Lambian: Yes, the prior distribution is ambiguous when the parameter space is infinite, as it is in this case: λ can assume an infinite number of values. When the parameter space is finite, however, the prior distribution is trivial: each of the possible values of the parameter have a priori the same weight. The poisson distribution is the limiting case of (hypergeometric) distributions with finite parameter spaces. You must compute first and take the limit later, rather than take the limit first and compute afterwards. (For example: a limit for big values of n is lim((1/n)/(1/n)) = lim(1) = 1, while the ratio lim(1/n)/lim(1/n) = 0/0 is undefined). There is no distribution giving each positive real the same weight, but there are distributions giving a finite number of reals the same weight, and when such distributions are used as prior, you may compute the posterior, and then take the limit. Note that the poisson distribution and the gamma distribution have the same form: ${\frac {\lambda ^{k}e^{-\lambda }}{k!}}.$ Considered a function of nonnegative integer values of k it is the discrete poisson distribution function, while considered a function of nonnegative real values of λ is is the continuous gamma distribution density function. If you sum over k you get 1, and if you integrate over λ you get 1. If you multiply by k and sum over k you get the mean value of the poisson distribution, which is λ , and if you multiply by λ and integrate over λ you get the mean value of the gamma distribution, which is k+1. Bo Jacoby 21:54, 23 September 2007 (UTC).

Thank you for the reply. It was very helpful.

Mark W. Miller 16:33, 24 September 2007 (UTC)

My pleasure! Bo Jacoby 21:38, 24 September 2007 (UTC).

Sum of factors

I know that the sum of the factors of 124 is 100 (1 + 2 + 4 + 31 +62). How would I go about finding all of the other numbers (say up to 10,000 or any arbitrary number) whose factors add to a specific number such as 100? 68.231.151.161 23:08, 22 September 2007 (UTC)

Write a computer program to factor and check all of the possible numbers. Offhand I'm not sure where the upper bound for possible candidates would be, but I'd guess around (2n)²? There is a Wikipedia article on Integer factorization, but depending on how big your numbers are, you might not need anything more complicated than trial division. (How come you left out 124 from the list of factors but kept 1?) - Rainwarrior 01:56, 23 September 2007 (UTC)

Actually, I have no idea if there is an upper bound, though under 80000 (which didn't take long to test with a computer program, actually, maybe a minute if I output all the factorizations to the screen) I only found one other number that matches your first example (194 -> 1 + 2 + 97 = 100). - Rainwarrior 02:24, 23 September 2007 (UTC)

Divisor function#Properties can be useful. If you are only interested in numbers with a specific sum then you can try working backwards from the sum formula to find possible prime factorizations. PrimeHunter 02:21, 23 September 2007 (UTC)

Actually, yeah, that's a good idea. For each set of monotonically increasing numbers that sums to 100, try to find a number with that list as a factorization (at worst you'd have to test every combination of numbers in the set multiplied with each other). That could make the search space a lot smaller. (And... my earlier upper bound estimate might be valid then... I was thinking prime factorization at first.) - Rainwarrior 02:56, 23 September 2007 (UTC)

I haven't proved this, but experimentally it appears to be the case that if the sum is s > 1, then the number is at most (s–1)², with equality when s = p+1, p a prime number.  --Lambiam 15:06, 23 September 2007 (UTC)

Well, if $n>(s-1)^{2}$ is prime, then its sum of factors is $1<s$, and if not, then it must have a factor greater than $s-1$, so adding 1 to that gives a sum greater than s. -- Meni Rosenfeld (talk) 16:23, 23 September 2007 (UTC)

I agree with PrimeHunter. Do you know how the sum of factors of a number can be written in terms of its prime factorization? Write 100 as the product of integers (greater than 1) in all possible ways, then see how each of those factors can be written as (1+p) or (1+p+p^2) or (1+p+p^2+p^3) etc for some prime p, and work backwards from there. – b_jonas 17:28, 24 September 2007 (UTC)

I don't understand the original assertion "...the factors of 124...(1 + 2 + 4 + 31 +62)". I can see including 2 & 62 in that list, and I can see including 4 & 31 in that list. However, the only reason that you can include 1 in that list is that it gives you 124 when you multiply it by 124. Thus, don't you have to include 124 in the list of factors, too? Or, what would make more sense to me, simply not include the identity factors in your list. Is this a definition thing, or an assumption that I don't see? It looks to me as though the list of factors is either (1,2,4,31,62,124) or (2,4,31,62). -SandyJax 18:41, 25 September 2007 (UTC)

Naturally, 1 is a factor of all numbers, because A*1 = A, no matter what A is. That might make it seem a trivial case, but it's still a factor that you can't validly exclude. But I take your point that 124 is also a factor of 124. We're told that factorization is "the decomposition of an object (for example, a number, a polynomial, or a matrix) into a product of other objects, or factors, which when multiplied together give the original. For example, the number 15 factors into primes as 3 × 5, and the polynomial x2 − 4 factors as (x − 2)(x + 2). In all cases, a product of simpler objects is obtained." I've highlighted the word "simpler". That suggests that a number is not a factor of itself. But that's a paradox if we accept that 1 is always a factor. Something's wrong somewhere. -- JackofOz 00:33, 26 September 2007 (UTC)

There's no deep mathematical issue here. It's just that "factor" isn't really a well-defined term, but more of a notion representing an idea. In the context of integers, we can take it to mean "divisor". Under this assumption, 124 is a factor of itself - but anon has, for whatever reason, chosen to consider the sum of only divisors smaller than the number. This is a pity, since the sum of all divisors is much easier to handle. -- Meni Rosenfeld (talk) 07:31, 26 September 2007 (UTC)