What Are Probability Axioms

One strategy in mathematics is to start with a few statements, then build up more mathematics from these statements. The beginning statements are known as axioms. An axiom is typically something that is mathematically self-evident. From a relatively short list of axioms, deductive logic is used to prove other statements, called theorems or propositions. The area of mathematics known as probability is no different. Probability can be reduced to three axioms. This was first done by the mathematician Andrei Kolmogorov. The handful of axioms that are underlying probability can be used to deduce all sorts of results. But what are these probability axioms? Definitions and Preliminaries In order to understand the axioms for probability, we must first discuss some basic definitions. We suppose that we have a set of outcomes called the sample space S.  This sample space can be thought of as the universal set for the situation that we are studying. The sample space is comprised of subsets called events E1, E2, . . ., En.   We also assume that there is a way of assigning a probability to any event E. This can be thought of as a function that has a set for an input, and a real number as an output. The probability of the event E is denoted by P(E). Axiom One The first axiom of probability is that the probability of any event is a nonnegative real number. This means that the smallest that a probability can ever be is zero and that it cannot be infinite. The set of numbers that we may use are real numbers. This refers to both rational numbers, also known as fractions, and irrational numbers that cannot be written as fractions. One thing to note is that this axiom says nothing about how large the probability of an event can be. The axiom does eliminate the possibility of negative probabilities. It reflects the notion that smallest probability, reserved for impossible events, is zero. Axiom Two The second axiom of probability is that the probability of the entire sample space is one. Symbolically we write P(S) 1. Implicit in this axiom is the notion that the sample space is everything possible for our probability experiment and that there are no events outside of the sample space. By itself, this axiom does not set an upper limit on the probabilities of events that are not the entire sample space. It does reflect that something with absolute certainty has a probability of 100%. Axiom Three The third axiom of probability deals with mutually exclusive events. If E1 and E2 are mutually exclusive, meaning that they have an empty intersection and we use U to denote the union, then P(E1 U E2 ) P(E1) P(E2). The axiom actually covers the situation with several (even countably infinite) events, every pair of which are mutually exclusive. As long as this occurs, the probability of the union of the events is the same as the sum of the probabilities: P(E1 U E2 U . . . U En ) P(E1) P(E2) . . . En Although this third axiom might not appear that useful, we will see that combined with the other two axioms it is quite powerful indeed. Axiom Applications The three axioms set an upper bound for the probability of any event. We denote the complement of the event E by EC. From set theory, E and EC have an empty intersection and are mutually exclusive. Furthermore E U EC S, the entire sample space. These facts, combined with the axioms give us: 1 P(S) P(E U EC) P(E) P(EC) . We rearrange the above equation and see that P(E) 1 - P(EC). Since we know that probabilities must be nonnegative, we now have that an upper bound for the probability of any event is 1. By rearranging the formula again we have P(EC) 1 - P(E). We also can deduce from this formula that the probability of an event not occurring is one minus the probability that it does occur. The above equation also provides us a way to calculate the probability of the impossible event, denoted by the empty set. To see this, recall that the empty set is the complement of the universal set, in this case SC. Since 1 P(S) P(SC) 1 P(SC), by algebra we have P(SC) 0. Further Applications The above are just a couple of examples of properties that can be proved directly from the axioms. There are many more results in probability. But all of these theorems are logical extensions from the three axioms of probability.

Data Warehousing Concepts, Products And Applications

The text book Data Warehousing concepts, techniques, products and applications by C.S.R. Prabhu. Mainly, the text book gives the information about the data model, online analytical processing systems and tools, data warehouse architecture, data mining algorithms, organizational issues of the data warehouse, data warehouse segmentation, Application of data mining and data warehousing. Firstly, the book describes Data Warehouse is a system where it is used for reporting the data from the wide range of the sources and indeed it helps the company to guide the management decisions. Moreover, Data Warehousing is the process where it evolved with the transformation and extraction of data from the various applications. Identically, it also has a technique from the formulation of the business intelligence where it gives effective implementation which makes the Data warehouse the effective technology for the business use. Importantly, Data Warehouse is the division of data into the individual data component. Similarly, Data Warehouse helps to analyze the data and whereas they are technologies which helps to analyze the data available in the data warehouse. Indeed, the functions of the Data Warehouse tools are data extraction, data cleaning, data transformation. Mainly, the data extraction gathers the data from the multiple sources, data cleaning helps to find and correct errors in data, data transformation converts the data into data warehouse format. Consequently, data cleaning andShow MoreRelatedAn Overview Of Data Warehousing1707 Words   |  7 PagesOverview of Data Warehousing Samuel Eda Wilmington University Abstract Data warehousing is a crucial element of decision supporting process, which now for a long time has become a focus of the database industry. 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Capital Punishment Is Always Wrong, Do You Agree Free Essays

Capital Punishment is always wrong, do you agree? A Christian would agree as they believe all life is sacred and only god can choose when your life ends, he will punish you in hell for your sins and it is not the place of the government to decide whether you are killed or not. This is because in the bible it says â€Å"Do not kill† which is one of the Ten Commandments which are not to be disobeyed, they also believe in forgiveness and love which are the fundamentals of their religion therefore criminals should be forgiven and given a chance to change. A Buddhist would agree because they believe in the sanctity of life. We will write a custom essay sample on Capital Punishment Is Always Wrong, Do You Agree? or any similar topic only for you Order Now This is because they believe in karma – that you will accumulate bad karma for the bad things you do in your life and will pay for them in your next life and so therefore we should leave people to be punished in their next life and not kill them because all life is sacred and if we kill them we will accumulate bad karma. A Muslim would disagree because they believe that punishment is needed to protect the welfare of society and serve justice to the victims family, they believe that if you take a life then you should die because you have taken away something sacred and sinned, this is because in the Qur’an it says â€Å"take not life, which God hath made sacred, except by way of justice and law† which permits the death penalty if it is to bring about justice. Another Christian would disagree because they believe that god has appointed the king/queen who has then appointed the government to serve justice on his behalf, this means that they can punish people in the means they feel appropriate and in the bible there are examples where people were killed for their sins so therefore it must be alright to kill people in today’s society. This is because in the bible it says â€Å"show no pity, life for life, eye for eye, tooth for tooth, hand for hand, foot for foot† which shows that we should have no mercy and should give the same punishment as the crime done for example if someone kills then they should be killed. I disagree as I believe sometimes it could be right because some criminals never reform and in killing them we would be preventing them from killing again and possibly deterring other criminals as they would be afraid of the punishment they would receive. How to cite Capital Punishment Is Always Wrong, Do You Agree?, Essay examples

The Catcher In The Rye Essays (718 words) - Literary Realism

The Catcher In The Rye The Catcher in the Rye In J.D. Salinger's The Catcher in the Rye, the first person narration is critical in helping the reader to know and understand the main character, Holden Caulfield. Holden, in his narration, relates a flashback of a significant period of his life, three days and nights on his own in New York City. Through his narration, Holden discloses to the reader his innermost thoughts and feelings. He thus provides the reader not only with information of what occurred, but also how he felt about what happened. Holden's thoughts and ideas reveal many of his character traits. One late Saturday night, four days before the beginning of school vacation, Holden is alone, bored and restless, wondering what to do. He decides to leave Pencey, his school, at once and travels to New York by train. He decides that, once in New York, he will stay in a cheap motel until Wednesday, when he is to return home. His plan shows the reader how very impetuous he is and how he acts on a whim. He is unrealistic, thinking that he has a foolproof plan, even though the extent of his plans are to take a room in a hotel.., and just take it easy till Wednesday. Holden's excessive thoughts on death are not typical of most adolescents. His near obsession with death might come from having experienced two deaths in his early life. He constantly dwells on Allie, his brother's, death. From Holden's thoughts, it is obvious that he loves and misses Allie. In order to hold on to his brother and to minimize the pain of his loss, Holden brings Allie's baseball mitt along with him where ever he goes. The mitt has additional meaning and significance for Holden because Allie had written poetry, which Holden reads, on the baseball mitt. Holden's preoccupation with death can be seen in his contemplation of a dead classmate, James Castle. It tells the reader something about Holden that he lends his turtleneck sweater to this classmate, with whom he is not at all close. Holden's feelings about people reveal more of his positive traits. He constantly calls people phonies, even his brother, D.B., who has sold out to Hollywood. Although insulting, his seemingly negative feelings show that Holden is a thinking and analyzing, outspoken individual who values honesty and sincerity. He is unimpressed with people who try to look good in other's eyes. Therefore, since it is obvious that Holden is bright, the reason for his flunking out of school would seem to be from a lack of interest. Holden has strong feelings of love towards children as evidenced through his caring for Phoebe, his little sister. He is protective of her, erasing bad words from the walls in her school and in a museum, in order that she not learn from the graffiti. His fondness for children can be inferred when he tells her that, at some time in the future, he wants to be the only grown-up with all these little kids playing some game in this big field of rye and all. He'll stand on the edge of a cliff and catch anybody who starts to fall off the edge of the cliff. He got this image from his misinterpretation of a line from the Robert Burns poem, if a body catch a body comin' through the rye. When situations are described, in person or in a book, they are influenced by the one who describes them, and by his or her perceptions and experiences. Through Holden's expressions of his thoughts and feelings, the reader sees a youth, sensitive to his surroundings, who chooses to deal with life in unique ways. Holden is candid, spontaneous, analytical, thoughtful, and sensitive, as evidenced by his narration. Like most adolescents, feelings about people and relationships are often on his mind. Unfortunately, in Holden's case, he seems to expect the worst, believing that the result of getting close to people is pain. Pain when others reject you or pain when they leave you, such as when a friend walks off or a beloved brother dies. It would not have been possible to feel Holden's feelings or understand his thoughts nearly

Essay on SELLING THE PASS HABEAS

Essay on SELLING THE PASS HABEAS Essay on SELLING THE PASS HABEAS CURRENT DEVELOPMENTS PUBLIC INTERNATIONAL LAW Edited by Dominic McGoldrick and Sarah Williams SELLING THE PASS: HABEAS CORPUS, DIPLOMATIC RELATIONS AND THE PROTECTION OF LIBERTY AND SECURITY OF PERSONS DETAINED ABROAD Abstract On 31 October 2012 the Supreme Court of England and Wales handed down its judgment in Rahmatullah v Secretary of State for Foreign Aï ¬â‚¬airs and Secretary of State for Defence [2012] UKSC 48. The case concerns an application for habeas corpus brought by a citizen of Pakistan originally detained by the United Kingdom in Iraq before being transferred into the custody of the United States. Rahmatullah addresses important issues concerning the extraterritorial reach of habeas corpus under English law in respect of persons held in the custody of a foreign State, as well as the international rule of law. The case may be considered a legal victory for persons detained without trial by the US in facilities thought to be beyond the reach of the courts. However, in reality any strength in the arm of the law is drained by the priority given to the conduct of foreign aï ¬â‚¬airs, ‘forbidden territory’ for the courts, over the Court’s ruling and the UK’s obligations under international law. The case is examined in the light of similar jurisprudence from US and Australian courts. Keywords: diplomatic relations, foreign nationals, habeas corpus, international law. I. INTRODUCTION On 31 October 2012 the Supreme Court of England and Wales handed down its judgment in Rahmatullah v Secretary of State for Foreign Aï ¬â‚¬airs and Secretary of State for Defence.1 The case concerns whether a writ of habeas corpus was properly issued against the Secretaries of State with regard to a non-British citizen detained by the United States (US) authorities at Bagram airbase, Afghanistan.2 Mr Rahmatullah (R) had been arrested by the United Kingdom (UK) armed forces in Iraq in 2003 and was subsequently transferred to US custody pursuant to a Memorandum of Understanding (MoU) between the UK and the US.3 The Court held unanimously that there was 1 Rahmatullah v Secretary of State for Foreign Aï ¬â‚¬airs and Secretary of State for Defence [2012] 3 WLR 1087 (SC(E)) (‘Rahmatullah’). 2 At the time of writing Mr Ramatullah remains in US detention. 3 Rahmatullah (n 1) para 3. The MoU is entitled, ‘An Arrangement for the Transfer of Prisoners of War, Civilian Internees, and Civilian Detainees between the Forces of the United States of America, the United Kingdom of Great Britain and Northern Ireland, and Australia, 23 March 2003.’ The original MoU is unavailable to the author. [ICLQ vol 62, July 2013 pp 727–739] doi:10.1017/S0020589313000225 728 International and Comparative Law Quarterly suï ¬Æ'cient uncertainty as to whether the UK retained control over R’s detention for the court properly to issue the writ and that there was no reason to believe, at the time the writ was issued, that it would not be eï ¬â‚¬ective. However the majority4 held that the Secretaries of State had fulï ¬ lled their duties pursuant to the writ by providing evidence of the US’s refusal to return R upon their request. The writ of habeas corpus sits at the heart of constitutional law.5 It is fundamental to protect the liberty of the individual from the unlawful use of executive power.6 If the detention of a person cannot be legally justiï ¬ ed, they have the right to be released. The court has no discretion to consider matters of public policy.7 In the words of Lord Carnwath and Lady Hale in this case, ‘[t]he strength of habeas corpus is its simplicity’.8 As Lord Kerr explained in the leading judgment, it is a ï ¬â€šexible remedy that has developed over the centuries so that it may be directed not only to the authority that has actual physical custody of the detainee but also to the person or authority that has a reasonable prospect of securing control over the detainee or his production to the court.9 This principle applies whether the detainee is within the jurisdiction of the court or has been

How to Stay Organized in College

How to Stay Organized in College You might have had grand plans about getting organized in college. And yet, despite your best intentions, your plans for organization seemed to slip through your fingers. So how can you stay organized for the long road ahead? Fortunately, although there are a zillion things to manage between your first day of classes and your last, staying organized in college is a lot easier than you might think. With a little advanced planning and the right skill set, staying organized can become more of your routine instead of just your ideal. Try Various Time Management Systems If you were wholly dedicated to making some fancy schmancy new calendaring app work for you this semester, but it ended up not working at all, dont be too hard on yourself. That means a particular system didnt work for you, not that youre bad at time management. Keep trying (and trying and trying) new time management systems until you find one that clicks. And if that means using a good, old-fashioned paper calendaring system, so be it. Having some calendar is the most important part of staying organized through the chaos that is college. Keep Your Dorm Room Clean When you lived at home, you had to keep your room relatively clean. But now that youre in college, you can keep your dorm room as messy as you want, right? Wrong! As silly as it sounds, a messy dorm room can represent a messy college life. Keeping your living space clean can help with everything from preventing you from losing your keys (again) to being able to mentally focus when you need to since you wont be visually distracted by all the junk on your desk. Additionally, keeping your space clean doesnt have to take a lot of time and will lead to all those little things that make you feel like youre in control of your own life: having clean clothes to choose from in the morning, knowing where that FAFSA form went, always having your cell phone charged. If keeping your dorm room clean seems like a waste of time, spend one-week tracking how much time you spend maintaining it clean and another week tracking how much time you spend looking for stuff or trying to recover from things youve lost (like that FAFSA form). You might surprise yourself. Stay on Top of Your Responsibilities When you are faced with anything that connects with your college life responsibilities - from a cell phone bill to an email from your mom about when youre coming home for Thanksgiving - make yourself do one of four things: Do it Schedule it Toss itFile it As an example, spending the next month arguing with your mom over when youll fly home will take ten times as much time as it will for you to just give her some dates when she brings it up. And if you arent sure yet, figure out a day by which you will be sure - and then put it into your calendaring system. Your mom will leave you alone, youll knock something off your to-do list, and you wont have to spend time telling yourself Oh shoot, I need to figure out Thanksgiving a million times a day between now and then. Spend Time Each Week Reorganizing Youre in college because youve got a great brain. So put it to use on all you have to do outside of the classroom! Just like a finely tuned athlete, your mind is learning, expanding, and strengthening each week; youre in school. Consequently, what organizing systems worked for you a month or two ago might not work any longer. Spend a few moments looking at what you got done, what youre doing, and what youll need to do over the next few weeks. While it may seem like a waste of time, those precious minutes can save you a lot of lost time - and a lot of disorganization - in the future. Plan Ahead to Stay Ahead Everyone knows that student who always says, Oh, I cant do something then, Ill be up all night cramming for my midterm. Really? Because that is just planning to be disorganized! Plan for everything you have to do. If you have a significant event youre planning, make sure your homework is done ahead of time so you can focus on your event when the time comes. If you know you have a major paper due, plan to work on it - and finish it - a few days in advance. Since its on your calendar and in your master plan, youll stay organized and on top of your tasks without even having to think about it. Take Care of Your Physical, Emotional, and Mental Health Being in college is hard - and not just academically. If youre not eating healthy, getting enough sleep, finding time to exercise, and overall treating yourself kindly, it will catch up with you sooner or later. And its impossible to get and stay organized if you dont have the physical, emotional, and mental energy to function. So give yourself a little TLC and remember that taking care of your health is an integral part of reaching your college goals.

Letter of Recommendation for a Professor at my school Essay

Letter of Recommendation for a Professor at my school - Essay Example Dr. Everett Shock is also such an engaging teacher that he/she is able to present challenging problems that made us look forward to solve the same and how it is applied in the real world setting. Dr. Everett Shock love for knowledge is contagious. His/her enthusiasm about the subject matter rubbed on us that made everybody look forward for the next class. His/her method of teaching does not only elucidate, but also encourage the students to excel thus attracting the best and brightest people to be better. Dr. Everett Shock is an authority on the subject evident with his many publications in scientific journals. But despite of his academic stature, he/she is very generous to accommodate students in his/her research that allows us to grow in ways that classroom experiences cannot afford. This was evident with my interaction with Dr. Shock’s Geo-Pig lab group in these past two years which was my first undergraduate lab research experience. The experience took me to an adventure in Yellowstone National Park for two summers with Dr. Shock and his research group and work side by side in the field learning about natural systems. The experience was priceless because it did only help me build a resume, but also enable me to have lasting friendship with very interesting people in the labs. This experience was not just exclusive to me, but also with many budding students who wanted to have first-hand experience in the labs which Dr. Shocks also enthusiastically accommodate to satisfy our eagerness to learn more. Such is the kind of teacher that Dr. Everett Shock is. Dr. Shock does not only encourage us to learn about theoretical concepts but also the experimental aspects of the science and its application to the real world. For example, I have always used thermodynamic parameters printed in the back of chemistry textbooks for calculating the free energy changes or the combustion energies of chemical reactions without giving much