The central limit theorem

The central limit theorem The Central Limit Theorem The central limit theorem is the second fundamental theorem in probability after the ‘law of large numbers. The‘law of large numbersis atheoremthat describes the result of performing the same experiment a large number of times. According to the law, theaverageof the results obtained after a large number of trials should be close to theexpected value, and will tend to become closer to this value as more trials are carried out. For example, a single roll of afair diceproduces one of the numbers {1, 2, 3, 4, 5, 6} each with equalprobability. Therefore, the expected value (E(x)), of a single dice roll is (1+2+3+4+5+6) à · 6 = 3.5. If this dice is rolled a large number of times, the law of large numbers states average of the result of all these trials known as the sample mean , will be approximately equal to 3.5. = 1Nk=1Nxk≈Ex=3.5 If the number of trials was to further increase, the average would further approach the expected value. So in general, as N→∞, →Ex This is the main premise of the law of large numbers. The central limit theorem is similar to the law of large numbers in that it involves the behaviour of a distribution as N→∞. The central limit theorem states that given a distribution with a mean (ÃŽ ¼) and variance (ÏÆ' ²), the samplingdistribution of the mean approaches anormal distributionwith a mean (ÃŽ ¼) and a variance (ÏÆ' ²N) as N, thesample size,increases. In other words, the central limit theorem predicts that regardless of the distribution of the parent population: Themeanof the population of means isalwaysequal to the mean of the parent population from which the population samples were drawn. Thestandard deviationof the population of means is always equal to the standard deviation of the parent population divided by the square root of the sample size (N). Thedistribution of means will increasingly approximate anormal distributionas the size N of samples increases. →X~N(ÃŽ ¼, ÏÆ'2N) (This is the main consequence of the theorem.) The origin of this celebrated theorem is said to have come from Abraham de Moivre, a French born mathematician who used the normal distribution to approximate the distribution of the number of heads resulting from many tosses of a fair coin. This was documented in his book ‘The Doctrine of Chances published in 1733 which was essentially a handbook for gamblers. This finding was somewhat forgotten until the famous French mathematicianPierre-Simon Laplacerevived it in his monumental work‘Thà ©orie Analytique des Probabilità ©s, which was published in 1812. Laplace was able to expand on de Moivres findings by approximating the binomial distribution with the normal distribution. De Moivre Laplace But as with de Moivre, Laplaces finding received little attention in his own time. It was not until the nineteenth century was at an end that the importance of the central limit theorem was discerned, when, in 1901, Russian mathematicianAleksandr Lyapunovdefined it in general terms and proved precisely how it worked mathematically.A full proof of the central limit theorem will be given later in this document. One may be familiar with the normal distribution and the famous ‘bell shaped curve that is associated with it. This curve is often found when presenting data for something like the heights or weights of people in a large population. Where ÃŽ ¼ is the mean . When the central limit theorem is applied, the distribution will approach something similar to the graph above. However, the amazing implication The central limit theorem explains why many non-normal distributions tend towards the normal distribution as the sample size N increases. This includes uniform, triangular, inverse and even parabolic distributions. The following illustrations show how they tends towards a normal distribution:

Reading Maketh a Full Man Essay

Did you know that there are nearly 800 million people around the world today who cannot read? All 320 of us can, but so many others can’t. And that’s when we realize just how important reading really is. Good morning, respected judges, teachers and dear friends. My name is Merlin Raphael and I’m here to talk about how reading makes a man. There was a time, without any screens, when reading was a popular leisure activity. People would spend hours reading and travelling to distant, exotic lands -in their minds, with only a leather bound book. But somewhere along the road, the passion to read is lost. And that’s just sad because reading offers a completely new life. Sir Francis Bacon, once said, and I quote, â€Å"Reading maketh a full man.† And he couldn’t be more correct. Reading involves a considerable level of concentration and thus, nurtures patience. Such an indulgence constantly elevates one’s vocabulary and knowledge. In fact, the habit can become a healthy addiction and helps us to stay in-touch with contemporary writers as well as those from the past. It also helps us to deal with issues of the modern world. Reading different types of texts make an even fuller man. What I mean is that he kind of knowledge you get from news articles is very different from reading stories. Being ignorant can be prevented with reading. It opens this huge wooden window and we finally start understanding other people and their lives. And a deeper awareness leads to tolerance and respect, creating a more civilized society. Furthermore, when one is caught up in a book, one comes across an array of emotions and somewhere in the middle; we find the sheer happiness of the reader. Lack of reading in one’s life can hinder gaining new knowledge, professional growth and the pure pleasure of reading. After all, no reading might even disturb one’s integrity. According to Confucius, we must find time for reading or surrender to â€Å"self-chosen ignorance†. You guys, reading is not merely a process of examining and understanding a text. There is a certain aura of finesse around one who reads. If a person can obtain so much from it, I think it’s safe to say that reading make a man.

The Effect of Exposure to Alcohol During Prenatal Development

Exposure to alcohol during prenatal development is one of the leading causes of preventable birth defects and mental impairments (Bower, Szajer, Mattson, Riley, Murphy, 2013; Nuà ±ez, Roussotte, Sowell, 2011; Sowell et al., 2008a). Fetal Alcohol Syndrome (FAS), the more severe form of PAE, affects two to seven out of every 1000 individuals in the United States (May et al., as cited in Simmons, Thomas, Levy Riley, 2010). The term Fetal Alcohol Spectrum Disorder (FASD) describes an individual with varying levels of PAE. Sampson et al. (as cited in Simmons at al., 2010) estimated that one in every 100 individuals suffers from some form of FASD. With such a high prevalence in the population, numerous studies have been done showing the†¦show more content†¦Alcohol has been shown to alter normal brain development by interfering with normal cell processes responsible for cell division and gene expression. Guerri and Renau-Piqueras (1997) explained that alcohol affects cell d ivision by inhibiting cytoskeleton activity. This increases the cells in the G0 and G1 phase, which reduces the number of mitotic cell undergoing cell division. A later study by Anthony, Zhou, Ogawa, Goodlett, and Ruiz (2008) found similar findings in the reduction of mitotic cells, but through induced apoptosis. They used mouse dorsal root ganglion stem cells to research the affect of PAE on the G1 and S phase of the cell cycle. Cell cultures were either exposure to 200 mg/dl or 400 mg/dl of alcohol, excluding the control group. These researchers found that alcohol exposure increased the levels of cyclin D1, cyclin D2, and E2F1, each of which play an important role in the regulation of the G1 and S phase of the cell cycle as well as subsequent cell division. Over expression of these proteins leads to decreased regulation in the cell cycle, altered DNA synthesis, increased DNA fragmentations, and ultimately apoptosis. This supports their findings of reduced cell numbers in the cultures exposed to alcohol. In addition, alcohol has been found to affect DNA replication by altering the DNA methylation process. DNA methylation (5mC) isShow MoreRelatedFetal Alcohol Syndrome And Its Effects1712 Words   |  7 Pages Fetal Alcohol Syndrome and its Effects On a Child’s Cognitive Development Dane D. Hrencher Kansas State University Fetal Alcohol Syndrome and its Effects On a Child’s Cognitive Development Alcohol is known as one the most dangerous teratogens. Every time a pregnant woman drinks, she allows alcohol to enter her blood stream and make its way into the placenta. 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Additionally, Chasnoff observed that children that had been exposed to alcohol during pregnancyRead MoreThe Effects Of Prenatal Alcohol On Children977 Words   |  4 Pagesindicated that prenatal exposure to high amounts of alcohol induced several developmental problems as the child grew up. It has been described that alcohol exposure during pregnancy causes several developmental problems in children. Some of these problems include a slower reaction to both visual and auditory stimuli (Mattson, 2006), growth retardation problems (Chasnoff, 2015), and motor impairment (Lucas, 2014). Additionally, Chasnoff observed that children that had been exposed to alcohol during pregnancyRead MorePrenatal Development And Development Of A Fetus1591 Words   |  7 PagesPrenatal development, also known as antenatal development, is the process of the development of a human fetus during pregnancy, from fertilization of the egg until the birth of the child. 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The Different Parts of a Stream

A stream is any body of running water that occupies a channel. It is normally above ground, eroding the land that it flows over and depositing sediment as it travels. A stream can, however, be located underground or even underneath a glacier.   While most of us speak of rivers, geoscientists tend to call everything a stream. The boundary between the two can get a little blurry, but in general, a  river  is a large surface stream. It is made up of many smaller rivers or streams. Streams smaller than rivers, roughly in order of size, may be called branches or forks, creeks, brooks, runnels, and rivulets. The very smallest kind of stream, just a trickle, is a rill. Characteristics of Streams Streams may be permanent or intermittent—occurring only part of the time. So you could say that the most important part of a stream is its channel or streambed, the natural passage or depression in the ground that holds the water. The channel is always there even if no water is running in it. The deepest part of the channel, the route taken by the last (or first) bit of water, is called the thalweg (TALL-vegg, from the German for valley way). The sides of the channel, along the edges of the stream, are its banks. A stream channel has a right bank and a left bank: you tell which is which by looking downstream. Stream channels have four different channel patterns, the shapes they show when viewed from above or on a map. The curviness of a channel is measured by its sinuosity, which is the ratio between the length of the thalweg and the distance downstream along the stream valley. Straight channels are linear or nearly so, with a sinuosity of nearly 1. Sinuous channels curve back and forth. Meandering channels curve very strongly, with a sinuosity of 1.5 or more (although sources differ on the exact number). Braided channels split and rejoin, like the braids in hair or a rope. The top end of a stream, where its flow begins, is its source. The bottom end is its mouth. In between, the stream flows through its main course or trunk. Streams gain their water through runoff, the combined input of water from the surface and subsurface. Understanding Stream Order Most streams are tributaries, meaning that they drain into other streams. An important concept in hydrology is stream order. A streams order is determined by the number of tributaries that flow into it.  First-order streams have no tributaries. Two first-order streams combine to make a second-order stream; two second-order streams combine to make a third-order stream, and so on.   For context, the Amazon River is a 12th order stream, the Nile an 11th, the Mississippi a tenth and the Ohio an eighth.   Together, the first through third-order tributaries making up the source of a river are known as its headwaters. These make up approximately 80% of all of the streams on Earth. Many large rivers divide as they near their mouths; those streams are distributaries. A river that meets the sea or a large lake may form a delta at its mouth: a triangle-shaped area of sediment with distributaries flowing across it. The area of water around a river mouth where seawater mixes with freshwater is called an estuary. Land Around a Stream The land around a stream is a valley. Valleys come in all sizes and have a variety of names, just like streams. The smallest streams, rills, run in tiny channels also called rills. Rivulets and runnels run in gullies. Brooks and creeks run in washes or ravines or arroyos or gulches as well as small valleys with other names. Rivers (large streams) have proper valleys, which may range from canyons to enormous flat lands like the Mississippi River Valley. The larger, deeper valleys are usually v-shaped. The depth and steepness of a river valley depends on the size, slope, and speed of the river as well as composition of the bedrock.   Edited by Brooks Mitchell