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June 1

Independence of bivariate normal distribution.

I found, that a bivariate normal distribution is independent, if it is uncorrelated. This appears the be the case when s12 and s21 are zero. Is there a proof with the steps taken for this? --193.170.229.238 (talk) 03:55, 1 June 2015 (UTC)[reply]

Calculation regarding WMF finances.

Consider the following:

If the WMF were to limit the growth of expenditures to some reasonable amount (I am thinking 5%, 10% at the most) starting now and we make some reasonable assumptions about future growth of revenue and future return on investments while keeping the principle safe, how long would it take for us to build up enough assets that we can fund everything from the interest and stop running banner ads asking for donations? How long would it take if we started by cutting expenditures to 2010 levels?

I did some back-of-the-envelope calculations, but I am seriously considering posting an RfC suggesting that we actually do this, so I would like to see what kind of numbers other people get.

I did a similar calculation a while back with the US budget and a rollback to 2000 expenditures (Bill Clinton). If anyone is interested I will try to find it. --Guy Macon (talk) 04:47, 1 June 2015 (UTC)[reply]

While this link is based on personal finance rather than business finance, the maths seems equally valid: [1]. A very quick glance at the graph shows that 2010 spending would be about 20% of current revenue, so if you could keep both of those up for 5.5 years, then the WMF could survive off the interest. Of course, unlike a household, the costs of wikipedia are likely to rise as it gets larger and more popular (as well as rises due to inflation), so the "safe withdrawal rate" for investments is probably somewhat lower. MChesterMC (talk) 08:49, 1 June 2015 (UTC)[reply]

Judging from the chart, WMF experiences exponential growth in both revenue and expenditures, not the usual case for personal finance. Usually such a curve follows the logistic model, which looks almost exactly like the exponential curve at the start but eventually levels off. So until there's a clear point of inflection visible it would be impossible to predict what the final limit would be. In any case, I doubt such a simple model could adequately predict future costs, not to mention "black swan" type events. --RDBury (talk) 14:01, 1 June 2015 (UTC)[reply]

Also, even if we somehow fix the growth models and ignore black swan events, the results are likely to be accutely sensitive to the assumed (1) rate of revenue growth, (2) rate of expenditure growth, and (3) rate of (real) returns on investment. And given that (roughly speaking) sum/differences of these parameters enter the equation in the exponent, small differences in the assumed values will lead to either wikimedia owning the world, or being bankrupt. Basically GIGO. Abecedare (talk) 14:34, 1 June 2015 (UTC)[reply]

"the costs of wikipedia are likely to rise as it gets larger and more popular" – I think the costs of running the Wikipedia servers are a tiny fraction of the WMF's current expenditures. Page views have less than doubled since 2008 and have been flat for several years now, but look at the WMF's spending in that time. -- BenRG (talk) 17:56, 1 June 2015 (UTC)[reply]

See my WP:BOLD proposal at [ https://meta.wikimedia.org/wiki/Talk:Wikimedia_Foundation_Annual_Plan/2015-16#A_simple_plan_to_solve_all_of_our_financial_problems_forever ]... :) --Guy Macon (talk) 03:10, 2 June 2015 (UTC)[reply]

Multiplication table - with numbers 1-10 & others

By zapping through other Wikipedias, it seems usual in many countries that pupils learn a multiplication table with the numbers 1-10. Is there a list / an analysis where this is the case, and where not? --KnightMove (talk) 06:17, 1 June 2015 (UTC)[reply]

Can you please clarify your question? Are you asking how commonly multiplication tables themselves are used in teaching arithmetic to children in different countries, or are you asking the common range for such educational multiplication tables? I was surprised by your 1-10 as I was taught 1-12 and had assumed everyone else was as well, though I recall being taught from flashcards, with my only recollection of multiplication tables being as interior cover decorations for notebooks. A quick Google image search on "multiplication table" returns 12x12 tables for 9 of the first 12 results (with the other three being 10x10, though the 13th result is 25x25), and our Multiplication table#History states, "The illustration below shows a table up to 12 × 12, which is a common size to teach in schools." Multiplication by 11 and 12 (and 10, for that matter) is not used in long multiplication (As Frank Mitchell wrote in his Mathematical Prodigies, "How many of us in multiplying 412,976 by 3,128, for instance, would think to treat the 12 in either the multiplicand or the multiplier as a single factor. It is only where the multiplier itself is simply 11 or 12 that the 12 x 12 table excels the 10 x 10 table for any practical purpose.") but multiplication by 11 is trivial and learning the 12-times table teaches the value of various number of dozens. There is some discussion of range (as well as a wonderful cartoon) in Talk:Multiplication table, and I would have expected to see this mentioned somewhere in mathematics education journals (particularly in regard to the debated value of rote memorization in teaching arithmetic), but I didn't find any. Sorry. -- ToE 12:52, 1 June 2015 (UTC)[reply]

ToE, I assume from your user name that you're of British origin? I was given to believe that we were taught our twelve times table so that we could handle our money, with 12d in the shilling, and that British schools now just go up to 10x10. Rojomoke (talk) 13:03, 1 June 2015 (UTC)[reply]

I'm just a Cranky Old Yank, though my primary education dates to the late '60s and early '70s at a small school in Pakistan with teachers hailing both from UK and elsewhere in the Commonwealth and from the US, so I should know better than to assume that my experience was typical of any other population. Your point about UK decimalization is an interesting one. I don't know about the interim, but I found several articles from 2012 regarding draft changes to the National Curriculum (England, Wales and Northern Ireland) such as this Telegraph article, "The comments come weeks after the Government published plans to overhaul the National Curriculum in England in a bid to promote the core knowledge that children should acquire at each age. A draft maths curriculum suggested that nine-year-old should know all their times tables up to 12x12 and confidently work with numbers up to 10 million by the end of primary school. Currently, children only need to know up to 10x10 and familiarise themselves with numbers below 1,000 by the age of 11." This apparently passed. Mathematics programmes of study: key stages 1 and 2 Pg. 17: "By the end of year 4, pupils should have memorised their multiplication tables up to and including the 12 multiplication table and show precision and fluency in their work." -- ToE 15:17, 1 June 2015 (UTC)[reply]

ToE, you have interpreted my question correctly. My motivation was primarily the English article Multiplication table, which does not mention the 10x10 table at all, and the introduction suggests a recommended table only up to 9x9?!. --KnightMove (talk) 14:48, 1 June 2015 (UTC)[reply]

While it looks like it's 12x12 in the UK, 9x9 may be all that is required from the Common Core State Standards Initiative in the US. Grade 3 » Operations & Algebraic Thinking: "Multiply and divide within 100. CCSS.Math.Content.3.OA.C.7: Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division (e.g., knowing that 8 × 5 = 40, one knows 40 ÷ 5 = 8) or properties of operations. By the end of Grade 3, know from memory all products of two one-digit numbers."

Common Core only sets minimum standards, and I would hope that most schools teach beyond this. While a 9x9 table might at first appear less intimidating than a 12x12 table, the x10 and x11 are trivial, and x12 are easy multiples of a dozen which need to be know for everyday life. For just a bit more work the student can feel a much greater sense of accomplishment, and, when expanding their understanding of arithmetic to multi-digit operations (Grade 4 » Number & Operations in Base Ten: "Use place value understanding and properties of operations to perform multi-digit arithmetic. CCSS.Math.Content.4.NBT.B.5: Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models."), the example 12 x 12 = (10 + 2) x (10 + 2) = 100 + 20 + 20 + 4 = 144, presented both algebraically and via the box method, will have greater impact when the student absolutely knows that 12 x 12 = 144 because it is a value that they memorized after first having constructed via repeated addition. The early wonder I found in mathematics came from the amazement that arithmetic could be computed in many different ways, but would yield the same value every time. -- ToE 16:23, 1 June 2015 (UTC) "What's eight times eight?" "Fifty-six," a tiger said. -- Richard Brautigan's In Watermelon Sugar[reply]

In Israel we use 1-10 (As can be evidenced by an image search for the Hebrew words for multiplication table). 1-12 seems completely bizarre to me. But then again, so does using a decimal number system with a non-decimal unit system. -- Meni Rosenfeld (talk) 16:14, 1 June 2015 (UTC)[reply]

I too learned 1-10. For multiplication I need 0-9, but 0 and 1 are trivial. So the 2-9 table is sufficient.

   */~2+i.8
 4  6  8 10 12 14 16 18
 6  9 12 15 18 21 24 27
 8 12 16 20 24 28 32 36
10 15 20 25 30 35 40 45
12 18 24 30 36 42 48 54
14 21 28 35 42 49 56 63
16 24 32 40 48 56 64 72
18 27 36 45 54 63 72 81

Bo Jacoby (talk) 06:56, 2 June 2015 (UTC).[reply]

And I guess Jacobi Jr. will only learn the J programming language? Count Iblis (talk) 20:40, 2 June 2015 (UTC)[reply]

In case anyone is interested, you really only need to know the table up to 5x5. Convert to balanced decimal, i.e, balanced ternary only with ten; the conversion can be done mentally. You can then to the conversion back to ordinary decimal in the addition phase. For example 729x183 = 1(-3)3(-1) * 2(-2)3 = 2(-6)6(-2)00 - 2(-6)6(-2)0 + 3(-9)9(-3) = 200000-80000+15000-1700+110-3=133407. True it's not really practical because is hard to keep track of all those minuses, but it is possible to to multiplication with a quarter of the memorization. Even more extreme, convert to binary-coded decimal, then you only have to memorize up to 1x1. --RDBury (talk) 01:18, 3 June 2015 (UTC)[reply]

Yes Count Iblis, so far J is the best programming language I have encountered. I like to get my result without much typing. The programming is done by 9 characters including the final carriage return. RDBury's nice transformation is: 729*183 = (700+20+9)(100+80+3) = (1000–300+30–1)(200–20+3). Bo Jacoby (talk) 10:02, 3 June 2015 (UTC). (Error fixed. Bo Jacoby (talk) 08:45, 4 June 2015 (UTC).)[reply]

Number of ways to arrange n items so that...

...the following criteria is met:

The items are numbered and no number occurs in its numerical order position.
If a sequence occurs where each item is followed by the one whose position matches the number of the preceding item, the sequence is periodic with n terms.

Examples:

a(1) = 0
a(2) = 1 (the only solution is 2,1)
a(3) = 2 (2,3,1 and 3,1,2)
a(4) = 6 (2,3,4,1; 2,4,1,3, 3,1,4,2, 3,4,2,1, 4,1,2,3, and 4,3,1,2)

What are a(5), a(6), and a(7)?? Note the reason, for example, 2,1,4,3 is not a valid way according to this rule because if we start with 2, the sequence would alternate 2,1,2,1,2,1,2,1,2,1,2,1,... contradicting the statement that the period should be 4 because the period is 2. Georgia guy (talk) 20:56, 1 June 2015 (UTC)[reply]

Your second criterion is unclear. I assume you mean by "The items are numbered" that if there are n items, they are consecutively numbered from 1 to n. But in the criterion it seems to the number of the item? From your example, it seems the latter. The numbers involved are small enough that a writing a computer program to exhaustively list and check each sequence is probably simplest. —Quondum 23:04, 1 June 2015 (UTC)[reply]

Pick a group of the numbers 1 to 5 so that no number occurs in its natural position and I'll tell you whether it qualifies. Georgia guy (talk) 23:34, 1 June 2015 (UTC)[reply]

A derangement is what a sequence satisfying the first condition is called. The second condition does not seem to be properly articulated. Sławomir Biały (talk) 23:48, 1 June 2015 (UTC)[reply]

Are you attempting to count the number of cyclic permutations without fixed points? -- ToE 00:53, 2 June 2015 (UTC)[reply]

Yes. For a=4, 3,1,4,2 is valid because the order will go 3,4,2,1,3,4,2,1,3,4,2,1, and so on. The period is 4, making it valid. But 2,1,4,3 is not valid because the order is 2,1,2,1,2,1,2,1,2,1, and so on. The period is 2, not 4. So it's invalid. Georgia guy (talk) 01:27, 2 June 2015 (UTC)[reply]

Then a(n) = (n-1)! for n>1. Check out Stirling numbers of the first kind. (a(1) is special because a 1-cycle is a fixed point.) -- ToE 01:34, 2 June 2015 (UTC)[reply]

You may also be interested in reading Permutation#Cycle notation. -- ToE 01:41, 2 June 2015 (UTC)[reply]