4/7/2013
The Central Limit Theorem
In the practice of statistics, most problems involving a significance test (z or t test), finding a probability, or the determination of a confidence interval requires the usage of normal approximations. Most populations have roughly normal distributions that allow for a random sample to be taken and tested without caution. However, there are some populations that just don’t fit the description of “normal”. They don’t follow a precise bell curve, and the lack of normality could leave a statistician wondering how they could possibly determine if normal approximations are appropriate. It is at this point that one of the most important theorems in statistical mathematics comes into play. The Central Limit Theorem, which was first introduced by Pierre-Simon Laplace in the late 18th century, states that as long as other important conditions are satisfied, a sample will become more normal as its size increases. An example of this is shown below. [pic]

As you can see, as “N” increased, the distribution began to appear more like a bell curve, which is just another way of saying “normal”.
Some rules have been applied to the Central Limit Theorem throughout its history that have contributed to it becoming one of the most consistently accurate mathematical theorems. For example, there are the previously mentioned ‘conditions’ that need to be met. In order to utilize the theorem, the sample must be taken randomly, pulled from the population, and it must show independence. The basic way to determine independence is through the 10% condition, which says that a sample is independent if it makes up less than 10% of the overall population. There is also the issue of deciding just how “large” the sample must be in order to be considered “large” enough to assume normality. For the most part, as long as the sample size (N) is greater than 30, the Central Limit Theorem can be applied. If N is between 10...

...central
The CentralLimitTheorem
A long standing problem of probability theory has been to find necessary and sufficient conditions for approximation of laws of sums of random variables. Then came Chebysheve, Liapounov and Markov and they came up with the centrallimittheorem. The centrallimittheorem allows you to measure the variability in your sample results by taking only one sample and it gives a pretty nice way to calculate the probabilities for the total , the average and the proportion based on your sample of information.
A statistical theory that states that given a sufficiently large sample size from a population with a finite level of variance, the mean of all samples from the same population will be approximately equal to the mean of the population. Furthermore, all of the samples will follow an approximate normal distribution pattern, with all variances being approximately equal to the variance of the population divided by each sample's size. Using the centrallimittheorem allows you to find probabilities for each of these sample statistics without having to sample a lot.
The centrallimittheorem is a major probability theorem that tells you what sampling distribution is used for many different statistics,...

...CENTRALLIMITTHEOREM
There are many situations in business where populations are distributed normally; however, this is not always the case. Some examples of distributions that aren’t normal are incomes in a region that are skewed to one side and if you need to are looking at people’s ages but need to break them down to for men and women. We need a way to look at the frequency distributions of these examples. We can find them by using theCentralLimitTheorem.
The CentralLimitTheorem states that random samples taken from a population will have a normal distribution as long as the sample size is sufficiently large. The sample mean will be approximately equal to the population mean. The sample’s standard deviation will be equal to the population’s standard deviation. The CentralLimitTheorem is so important because with it we will know the shape of the sampling distribution even though we may not know what the population distribution looks like.
The real key to this entire theorem is the term sufficiently large. If the sample size isn’t sufficiently large, the frequency distribution for the sample size will not look the same as it does for the population. For populations that are really symmetric, sample sizes of two or three will do. This is due to the fact that symmetric populations...

...Kazakhstan
24 september 2014
Geography of central Asia 24.09.2014
Central Asia is the core region of the Asian continent and stretches from the Caspian Sea in the west to China in the east and from Afghanistan in the south to Russia in the north. It is also sometimes referred to as Middle Asia, and, colloquially, "the 'stans" (as the six countries generally considered to be within the region all have names ending with the Persian suffix "-stan", meaning "land of")[1] and is within the scope of the wider Eurasian continent.
In modern contexts, all definitions of Central Asia include these five republics of the former Soviet Union: Kazakhstan (pop. 17.9 million), Kyrgyzstan (5.8 million), Tajikistan (8.0 million), Turkmenistan (5.2 million), and Uzbekistan (30.2 million), for a total population of 67.1 million as of 2013-2014. Afghanistan (pop. 31.1 million) is also sometimes included.
Central Asia is an extremely large region of varied geography, including high passes and mountains (Tian Shan), vast deserts (Kara Kum, Kyzyl Kum, Taklamakan), and especially treeless, grassy steppes.[16] The vast steppe areas of Central Asia are considered together with the steppes of Eastern Europe as a homogeneous geographical zone known as the Eurasian Steppe.
Much of the land of Central Asia is too dry or too rugged for farming. The Gobi desert extends from the foot of the Pamirs, to...

...Review of Finance (2010) 14: 157–187 doi: 10.1093/rof/rfp018 Advance Access publication: 4 October 2009
The Limits of the Limits of Arbitrage
ALON BRAV1 , J.B. HEATON2 and SI LI3
Professor of Finance, Duke University Fuqua School of Business; 2 Partner, Bartlit Beck Herman Palenchar & Scott LLP; 3 Assistant Professor of Finance, Wilfrid Laurier University School of Business and Economics Abstract. We test the limits of arbitrage argument for the survival of irrationality-induced financial anomalies by sorting securities on their individual residual variability as a proxy for idiosyncratic risk – a commonly asserted limit to arbitrage – and comparing the strength of anomalous returns in low versus high residual variability portfolios. We find no support for the limits of arbitrage argument to explain undervaluation anomalies (small value stocks, value stocks generally, recent winners, and positive earnings surprises) but strong support for the limits of arbitrage argument to explain overvaluation anomalies (small growth stocks, growth stocks generally, recent losers, and negative earnings surprises). Other tests also fail to support the limits of arbitrage argument for the survival of overvaluation anomalies and suggest that at least some of the factor premiums for size, book-to-market, and momentum are unrelated to irrationality protected by limits to...

...Richard C. Carrier, Ph.D.
“Bayes’ Theorem for Beginners: Formal Logic and Its Relevance to Historical Method — Adjunct Materials and Tutorial”
The Jesus Project Inaugural Conference “Sources of the Jesus Tradition: An Inquiry”
5-7 December 2008 (Amherst, NY)
Table of Contents for Enclosed Document
Handout Accompanying Oral Presentation of December 5...................................pp. 2-5 Adjunct Document Expanding on Oral Presentation.............................................pp. 6-26 Simple Tutorial in Bayes’ Theorem.......................................................................pp. 27-39
NOTE: A chapter of the same title was published in 2010 by Prometheus Press (in Sources of the Jesus Tradition: Separating History from Myth, ed. R. Joseph Hoffmann, 2010) discussing or referring to the contents of this online document. That primary document (to which this document is adjunct) has also been published in advance as “Bayes’ Theorem for Beginners: Formal Logic and Its Relevance to Historical Method” in Caesar: A Journal for the Critical Study of Religion and Human Values 3.1 (2009): 26-35.
1
Richard C. Carrier, Ph.D.
“Bayes Theorem for Beginners: Formal Logic and Its Relevance to Historical Method”
December 2008 (Amherst, NY) Notes and Bibliography 1. Essential Reading on “Historicity Criteria” Stanley Porter, The Criteria for Authenticity in Historical-Jesus Research: Previous...

...A Friend in Need is a Friend indeed Essay
There is nothing better than surrounded by good friends. You may look at some people and their friends with envy as they chat away happily and participate in activities together. It may be hard to figure out which friends are better when considering the friends who can have fun with and the friends that can get help from. From my perspective, the people who are willing to help me in the crisis time are much more cherished than who just want to stay with me to have fun.
First, it is easy to find abundant sources that make you entertaining and fun around you, but it is hard to find a true friend who can give you a hand when you are in need. What do you do when you are bored? You probably turn on the computer and play games or surf internet. Also, you can go outside play some sport. You do not need to have friend just for fun because there are many other activities to entertain you. However, what can you do, when you are in need of something that you cannot do it without some help from others? That is when you need a friend who can give you help when you need it. But when you need help, you need someone to help you. Like, in a school, you are absent one day, and teacher is going to discuss important materials for next test in class. No one really can help you to get that information even parents cannot do this for you. On the other hand, your good friend in your class can help you to have the information. He or she helps you to...

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Pythagorean Theorem
In mathematics, the Pythagorean theorem or Pythagoras' theorem is a relation in Euclidean geometry among the three sides of a right triangle (right-angled triangle). In terms of areas, it states:
In any right triangle, the area of the square whose side is the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares whose sides are the two legs (the two sides that meet at a right angle).
The theorem can be written as an equation relating the lengths of the sides a, b and c, often called the Pythagorean equation:[1]
where c represents the length of the hypotenuse, and a and b represent the lengths of the other two sides.
These two formulations show two fundamental aspects of this theorem: it is both a statement about areas and about lengths. Tobias Dantzig refers to these as areal and metric interpretations.[2][3] Some proofs of the theorem are based on one interpretation, some upon the other. Thus, Pythagoras' theorem stands with one foot in geometry and the other in algebra, a connection made clear originally byDescartes in his work La Géométrie, and extending today into other branches of mathematics.[4]
The Pythagorean theorem has been modified to apply outside its original domain. A number of these generalizations are described below, including...

...bernoulli's theorem
ABSTRACT / SUMMARY
The main purpose of this experiment is to investigate the validity of the Bernoulli equation when applied to the steady flow of water in a tape red duct and to measure the flow rate and both static and total pressure heads in a rigid convergent/divergent tube of known geometry for a range of steady flow rates. The apparatus used is Bernoulli’s Theorem Demonstration Apparatus, F1-15. In this experiment, the pressure difference taken is from h1- h5. The time to collect 3 L water in the tank was determined. Lastly the flow rate, velocity, dynamic head, and total head were calculated using the readings we got from the experiment and from the data given for both convergent and divergent flow. Based on the results taken, it has been analysed that the velocity of convergent flow is increasing, whereas the velocity of divergent flow is the opposite, whereby the velocity decreased, since the water flow from a narrow areato a wider area. Therefore, Bernoulli’s principle is valid for a steady flow in rigid convergent and divergent tube of known geometry for a range of steady flow rates, and the flow rates, static heads and total heads pressure are as well calculated. The experiment was completed and successfully conducted.
INTRODUCTION
In fluid dynamics, Bernoulli’s principle is best explained in the application that involves in viscid flow, whereby the speed of the moving fluid is increased...