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In probability theory, the expected value of a random variable is a key aspect of its probability distribution. The expected value of a discrete random variable is the probability-weighted average of all its possible values. In other words, each possible value the random variable can assume is multiplied by its probability of occurring, the resulting products are summed to produce the expected value. Intuitively, a random variable's expected value represents the mean of a large number of independent realizations of the random variable; the expected value of a probability distribution is known as the expectation, mathematical expectation, average, or first moment. Expected value applies to an continuous random variable, except that an integral of the variable with respect to its probability density replaces the sum; the formal definition includes both of these and works for distributions which are neither discrete nor continuous. The idea of the expected value originated in the middle of the 17th century from the study of the so-called problem of points, which seeks to divide the stakes in a fair way between two players who have to end their game before it's properly finished.

This problem had been debated for centuries, many conflicting proposals and solutions had been suggested over the years, when it was posed in 1654 to Blaise Pascal by French writer and amateur mathematician Chevalier de Méré. Méré claimed that this problem couldn't be solved and that it showed just how flawed mathematics was when it came to its application to the real world. Pascal, being a mathematician, was determined to solve the problem once and for all, he began to discuss the problem in a now famous series of letters to Pierre de Fermat. Soon enough they both independently came up with a solution, they solved the problem in different computational ways but their results were identical because their computations were based on the same fundamental principle. The principle is that the value of a future gain should be directly proportional to the chance of getting it; this principle seemed to have come to both of them. They were pleased by the fact that they had found the same solution and this in turn made them convinced they had solved the problem conclusively.

They only informed a small circle of mutual scientific friends in Paris about it. Three years in 1657, a Dutch mathematician Christiaan Huygens, who had just visited Paris, published a treatise "De ratiociniis in ludo aleæ" on probability theory. In this book he considered the problem of points and presented a solution based on the same principle as the solutions of Pascal and Fermat. Huygens extended the concept of expectation by adding rules for how to calculate expectations in more complicated situations than the original problem. In this sense this book can be seen as the first successful attempt at laying down the foundations of the theory of probability. In the foreword to his book, Huygens wrote: It should be said that for some time some of the best mathematicians of France have occupied themselves with this kind of calculus so that no one should attribute to me the honour of the first invention; this does not belong to me. But these savants, although they put each other to the test by proposing to each other many questions difficult to solve, have hidden their methods.

I have had therefore to examine and go for myself into this matter by beginning with the elements, it is impossible for me for this reason to affirm that I have started from the same principle. But I have found that my answers in many cases do not differ from theirs. Thus, Huygens learned about de Méré's Problem in 1655 during his visit to France. Neither Pascal nor Huygens used the term "expectation" in its modern sense. In particular, Huygens writes: That any one Chance or Expectation to win any thing is worth just such a Sum, as wou'd procure in the same Chance and Expectation at a fair Lay.... If I expect a or b, have an equal chance of gaining them, my Expectation is worth /2. More than a hundred years in 1814, Pierre-Simon Laplace published his tract "Théorie analytique des probabilités", where the concept of expected value was defined explicitly: … this advantage in the theory of chance is the product of the sum hoped for by the probability of obtaining it; this division is the only equitable one.

We will call this advantage mathematical hope. The use of the letter E to denote expected value goes back to W. A. Whitworth in 1901, who used a script E; the symbol has become popular since for English writers it meant "Expectation", for Germans "Erwartungswert", for Spanish "Esperanza matemática" and for French "Espérance mathématique". Let X be a random variable with a finite number of finite outcomes x 1, x 2, …, x k occurring with probabilities p 1, p 2

Warren Tyrone Harrell is a former Democratic member of the North Carolina General Assembly representing the state's 41st House district in western Wake County. He defeated Chris Mintz in the 2006 Democratic primary, incumbent J. Russell Capps in the 2006 general election. On September 20, 2009, just nine months into his second term in office, Harrell resigned from the North Carolina House of Representatives after separate investigations into his campaign expenditures were launched by the House Ethics Committee and the State Board of Elections. Harrell was graduated from Sanderson High School. Harrell received his B. A in English from Appalachian State University and his M. A in Political Management from The George Washington University, he is a member of the Kappa Alpha Psi fraternity. In early September 2009, the North Carolina State Board of Elections began an official audit of Harrell's campaign finance records, citing irregularities, unusual activity and incomplete entries. Among the items that caught the attention of auditors were hundreds of dollars in campaign expenditures at clothing and luggage stores marked as "committee meetings" on the paperwork Harrell filed.

Shortly after the Board of Elections' announcement, the NC House Speaker's Office announced that the Speaker had ordered an ethics investigation of Harrell over his financial records. On September 20, 2009, Harrell submitted a letter of resignation to House Speaker Joe Hackney, effective amid the ongoing controversy over campaign expenditures totaling more than \$13,000, revelations he was living outside of his district, his divorce from his wife. In his first term as Representative of the 41st district, Harrell supported measures for higher teacher and state employee salaries and quality healthcare, protection of local small businesses and collaborative university efforts in the search for renewable energy sources. Harrell began his second term as Chair of the House Committee on Science and Technology and vice-chair of the House Committee on State Government and State Personnel, as well as vice-chair of the House Appropriations Subcommittee on Transportation. Early in his second term, Harrell received criticism for his support of H. 1252 in his committee.

The bill was supported by various conservative organizations as well as Time Warner Cable, which had a location within then-Representative Harrell's district. The left-leaning North Carolina Center for Public Policy Research listed Harrell 52nd in their 2008 effectiveness rankings, the highest ranking given to any freshman lawmaker; those rankings have since been criticized for favoring the majority party, as they include votes from the media and the lawmakers themselves. In the group's subsequent 2010 rankings, the Raleigh News & Observer reported that "the dubious honor for the biggest drop in effectiveness belongs to former Rep. Ty Harrell, who dropped from 52nd to 110th." In June 2007, Harrell became the first elected official in North Carolina to endorse Barack Obama, after the president's election, Harrell was rumored to have been considered a potential choice as Obama's ambassador to Canada. NC House of Representatives - Ty Harrell official NC website DLCC- Representative Ty Harrell Project Vote Smart - Representative Ty Harrell profile Follow the Money - Ty Harrell campaign contributions: 2006 2008 The Return of Ty Harrell 2013

Dr John Anderson Gilruth was a veterinary scientist and administrator. He is noted for being Administrator of the Northern Territory from 1912 to 1918, when he was recalled after an angry mob demanded that he resign; this incident is known as the Darwin Rebellion. He was born in the son of Andrew Gilruth, he was educated at Arbroath High School and the High School of Dundee served two years as clerk to an Arbroath solicitor before going to Glasgow Veterinary College, now the Faculty of Veterinary Medicine at the University of Glasgow in 1887. He was admitted to membership of the Royal College of Veterinary Surgeons, London, in 1892, he accepted appointment as a government veterinary surgeon in New Zealand. In New Zealand from 1893, he spent three years investigating stock diseases a year at the Pasteur Institute in Paris. In 1896, on returning to New Zealand, he was appointed chief veterinarian and government bacteriologist, he was appointed a fellow of the Royal Society of Edinburgh in 1907. In 1908, he accepted the foundation chair of veterinary pathology at the University of Melbourne.

In 1907 he was elected a Fellow of the Royal Society of Edinburgh. His proposers were Frederick Hobday, Sir John McFadyean, John Berry Haycraft, Sir Edward Albert Sharpey-Schafer, he was married to Jennie McKay. Alcorta, F. X.. Darwin Rebellion. Northern Territory University Planning Authority, Darwin OCLC 27546680 Jensen, H. I.. The Darwin Rebellion. Labour History, no 11, November 1966, pp 3–13. Canberra. OCLC 84284994 Powell, Alan. "Gilruth, John Anderson". Australian Dictionary of Biography. Canberra: Australian National University. Retrieved 26 March 2008. John Anderson Gilruth. Northern Territory Government Administrators. Retrieved on 3 May 2008