Wikipedia:Reference desk/Archives/Mathematics/2016 July 14

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July 14

Recovering a group from a quotient

If I have a group G and a normal subgroup H of G, is there a natural, general, way to reconstruct G from knowledge of H and G/H? Taking the direct product seems to not work in cases such as $G=\mathbb {Z} /4\mathbb {Z} ,H=G/\{0,{\bar {2}}\}$. I'm also interested in seeing if there's an easy-to-apply sufficient condition for a direct product to work, for example when G has order p² for some prime p and is acyclic, with H the cyclic group of order p.--Jasper Deng (talk) 04:20, 14 July 2016 (UTC)

One way is the semidirect product, which covers more cases than the direct product, but still by no means all. An instructive example is to find the groups of order 8 having a cyclic normal subgroup of order 4. All but one of the five groups of order 8 meet this criterion, so knowing H and G/H actually tells you very little in this case. You can get a bit further in the case that H is contained in the center of G, and there is a whole theory of these central extensions. The general case is covered by the theory of group extensions, but it's probably best not to try to tackle it without a good understanding of of semidirect products and central extensions. --RDBury (talk) 06:35, 14 July 2016 (UTC)

Warren Buffett: luck or ability?

Could Warren Buffet's Berkshire Hathaway's success be explained by luck alone? Or does beating the S&P for four decades imply ability?Llaanngg (talk) 00:25, 14 July 2016 (UTC)

• The above was posted on the Humanities Desk. I'm moving it here in the hope that someone can answer it sensibly from a point of view of regression analysis or some other appropriate statistical test. --69.159.60.163 (talk) 04:33, 14 July 2016 (UTC)

According to this Business Insider graph, between 1964 and 2014, the S&P has gone up about 2300%, while Berkshire Hathaway has risen 1,800,000%, even with a few years where it didn't outperform the S&P. I'd say that that's conclusive evidence for staggering expertise or a really good deal with the Devil. Clarityfiend (talk) 08:38, 14 July 2016 (UTC)

Buffett is an exponent of value investing, and "buy and hold", which is to say he purchases undervalued shares and waits for them to come good. He finances his investments with a large liquidity pool, for example he owns many insurance companies, which have large cash assets and deferred liability. 86.168.123.201 (talk) 10:19, 14 July 2016 (UTC)

This bit, even right, would be better suited at another desk. Hofhof (talk) 10:33, 14 July 2016 (UTC)

It is still possible to explain this by luck. Imagine that the stock market is a casino (not that difficult). Warren Buffet went to the roulette, bet repeatedly his whole stack at red and won 14.2 times in a row, although the average was to win just 4.5 times in a row. Given the existence of millions of investors, it seems to me that some would necessarily have that amount of luck. Hofhof (talk) 10:31, 14 July 2016 (UTC)

As it says at the top, this section was transferred from the Humanities desk, although some comment seems to have been lost in the transfer. In the past 50 years Buffett has had four losing years - 1974 (didn't we all), 1990, 1999 (he's not a fan of technology) and 2008 (unsurprisingly). This is a correlation with the stock market, not with anything which might happen in a casino. The phenomenon described above was exploited in a television programme. After a discussion about how the toss of a coin could be influenced the presenter segued to a sequence in which a coin came up heads ten times in a row. What he didn't say was that in getting that sequence the cameraman had discarded many others filmed over days, getting closer and closer until finally they got the one they were waiting for.

Investment and racing tipsters work the same way - different subscribers get different tips so they can truthfully point to someone who has got rich following their advice. Another television programme was based on this - a pool of viewers were independently approached, offered a free horseracing bet and invited to come back for another after their horse won. The participants were grouped, with a different selection being given to each group. Finally, two couples whose tips had all been successful were invited to the racecourse, where the trick was explained, the final bets were placed and they watched the race. 86.168.123.201 (talk) 11:18, 14 July 2016 (UTC)

• My apologies. I did delete one response when moving the question. It was a non-mathematical comment on a mathematical question, and therefore not useful in my opinion, but that's not a reason to delete someone else's contribution, and I did not intend to. It was a careless accident (probably I did copy-and-paste but copied too little) and, again, I apologize to Bus stop. Here's what he or she said:

Why would anyone think that could be explained by "luck alone"? The possibility exists that luck alone could explain this phenomenon, but is there anything that would lead us to think this? Is he bad with numbers? Is he not conversant in this topic area? Bus stop (talk) 02:51, 14 July 2016 (UTC)

--69.159.60.163 (talk) 06:24, 16 July 2016 (UTC)

Still, this has nothing to do with maths. The question certainly belongs here, your answers not. Hofhof (talk) 12:07, 14 July 2016 (UTC)

The maths aspect would be "how many tosses would it take, on average, for a coin to come up "heads" ten times in a row"? The answer, presumably, is 2¹⁰, but that doesn't answer the question "how many lesser sequences would come up, on average, before this sequence was achieved"? 86.168.123.201 (talk) 12:19, 14 July 2016 (UTC)

The maths question (or stats actually) is what the OP asked. If you pick stock at random, what's your chance of multiplying your investment by x? Hofhof (talk) 13:15, 14 July 2016 (UTC)

Nobody has yet been able to devise an algorithm which will predict future stock prices. The reason appears to be that the index is composed of thousands of individual stocks, each subject to unique influences. This is much like predicting the weather - the entire atmosphere has been divided up into cubes, the data for each of which is fed into a supercomputer, but a small change in the input results in a huge change in the forecast. This is the "butterfly effect" - apparently a butterfly flapping its wings in London can cause a large variation in the weather in China. All these are "chaotic systems", like the behaviour of the water gushing out of a tap after you turn it on. What you can say, however, is that in the long term the value of shares will far outpace bonds, which in turn will outpace cash investments. The reason ascribed to this is that shares are a stake in the economy, and thus will compensate for inflation and participate in wealth creation. 194.66.226.95 (talk) 15:07, 14 July 2016 (UTC)

It is worse than that. People are competing with each other and trying to outguess each other so any possible 'solution' just makes the problem more difficult as soon as it is known about. It is a nasty brew of game theory and human psychology. I think the pact with the devil possibility makes the most sense. :) Dmcq (talk) 15:35, 14 July 2016 (UTC)

It could be luck, but the difference in return is ginormous. Still, there is Archie Karas, who turned $50 into $40 million (and then back to nothing) in Vegas. Clarityfiend (talk) 22:52, 14 July 2016 (UTC)

With sufficiently many gamblers, anything happens. Take for example 1024 gamblers, starting with $1 each. They gamble against one another. After first round there are 512 gamblers each having $2. After second round there are 256 gamblers each having $4. And so on. After tenth round there is one winner having $1024. The winner's success is explained by luck alone and does not imply ability at all. Bo Jacoby (talk) 16:40, 15 July 2016 (UTC).

You're supposing that all the gamblers would gamble everything they have until one person gets all the money. But it might not be like that. For example, someone who won in the first round might keep a dollar back for later and only bet a dollar in the second round. 81.151.129.196 (talk) 16:59, 15 July 2016 (UTC)

As Bo says, the point is that "with sufficiently many gamblers, anything happens". So to answer the question, you need to know how many investors it would take before it becomes statisticaly probable that someone matches Buffett's level of success, and compare that to the number of real people involved. --69.159.60.163 (talk) 06:28, 16 July 2016 (UTC)

One way to approach this might be to get hold of a 1964 Stock Exchange Daily Official List and one from today (I presume it's still published) and work through, comparing the share prices of individual companies then and now. This might take quite a bit of work, since shares may have been split or consolidated, companies change their names or get taken over and shares are sometimes exchanged for shares in other companies. You can then say what percentage of survivors has appreciated by a given amount. It's very rare for a quoted company to go bust, be delisted or be taken private, so you should be able to get data on most companies. 86.176.84.174 (talk) 12:14, 16 July 2016 (UTC)

Consider a population of gamblers. Consider two events: A, that a randomly chosen gambler is immensely rich, and B, that the richest gambler is immensely rich. The probability of A is low, but the probability of B is high. You do not need to assume any ability to explain B. Bo Jacoby (talk) 09:25, 17 July 2016 (UTC).

From [1] investing in one stock at random gives a return of 5.47% with a standard deviation of nearly 14.68%. Any decent investor will spread the risk between different sectors and that gives a better return on average when multiplied out over the years than holding only a single stock would, but for the maximum we can assume high risk takers who just have one stock a year. So the question is what percentage of them will reach 1,800,000% in 50 years when the stock market went up 2300%? Well that's (1800000/2300)^(1/50) a year or 14.25% a year, nearly an average one standard deviation per year for 50 years. The average standard deviation after 50 years would be 1//sqrt(50) of that for one year so we're talking about seven standard deviations. If you look at standard deviation you'll see six standard deviation including both sides would occur in 1/500000000 times and seven times would require a vastly expanded population for the earth. So I guess that they have some skill at what they are doing, and of course a pact with the devil is also a likely possibility. Dmcq (talk) 10:02, 17 July 2016 (UTC)

The quoted numbers, a return of 5.47% with a standard deviation of nearly 14.68%, are not necessarily applicable to the future, even if the data have been extracted from the past. Financial crises have happened and may happen again. Bo Jacoby (talk) 18:13, 17 July 2016 (UTC).

There's loads of things wrong with it. The distribution is fat tailed, it varies by year that is just an average, Warren Buffett keeps stocks for a number of years rather than switching every year, one can just go on an on. Can you get a better figure easily though? A more accurate way of looking at what might be happening is if they are a half a standard deviation better than the average rather than a full standard deviation, then something like 1/3000 doing it would get the returns of Warren Buffett, which sounds much more believable. A mixture of skill and luck. Dmcq (talk) 21:47, 17 July 2016 (UTC)

Multivariate normal distribution

This article state right in the beginning: "One possible definition is that a random vector is said to be k-variate normally distributed if every linear combination of its k components has a univariate normal distribution." Does the theorem "If every linear combination of a k-dimensional random vector's k components has a univariate normal distribution, it is k-variate normally distributed." have a name? Could you point me to a textbook that discusses this? (Is there a more general theorem "The set of distributions of all linear combinations of a random vector's components completely determines the multivariate distribution of the random vector."?) -- 134.76.84.240 (talk) 14:24, 14 July 2016 (UTC)

Regarding your first question, this is a definition, not a theorem, so it doesn't have a name. Regarding your second question, the section "Definition" says that this is discussed in the article's reference 1, and reference 2 has a promising title and is online. Don't know about the generalization, though it strikes me that it may be tautologically true but not helpful outside of the normal case.Loraof (talk) 21:55, 14 July 2016 (UTC)

Relation of R and *R

*R is said to be R extension, which as well as infinities it contains infinitesimals that R doesn't contain. More specifically, there is a chunk of *R space that satisfies 0 < e < r. How does that work with 0 < r/2 < r ?

I'm not sure what you mean by 'how does that work' but a simple version of such a space can be got by using a pair (x,e) for the numbers and defining (x,e) < (y,f) if x<y or x=y and e<f. Then the first element of the pair can be equivalent to a normal number and the second an infinitesimal addition. Try that out and see how it works. Dmcq (talk) 15:46, 14 July 2016 (UTC)