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. Vast number of commercial products and various services has been available now, and all of the top notch database management system vendors now have offerings in this area. This paper provides an overview of history of data warehousing, the type of systems in data warehousing, focusing onRead MoreData Warehousing And Information Warehousing1493 Words   |  6 PagesData warehousing is a system that holds the data of an organization collected through various channels and the data is processed through various analytical tools to generate reports for the business users. This paper discusses the data warehouse concept along with the origin of the data warehouse and the current trends of data warehousing. Various steps involved in the development of the data warehousing project are discussed in this paper. This paper also lists out the challenges en countered whileRead MoreAssignment 3 Business Intelligence and Data Warehouses752 Words   |  4 PagesIntelligence and Data Warehouses Student’s name: Professor’s name: Course title: 1. Differences between the structures of a relational database optimized for online transactions versus a data warehouse optimized for processing and summarizing large amounts of data Data Warehouse is a database which is designed to process for query and analysis rather than for transaction processing, and it is usually contains historical data derived from transaction data, but can include data from other sourcesRead MoreWarehousing And Storage Of Warehousing1724 Words   |  7 Pages2.1 Warehousing and storage Warehousing is a basic part of each logistics framework. Efficient and effective warehousing administration plays an important role in giving great customer service for less cost. The health section, through its diverse provincial health centres cannot sidestep warehousing in giving health care. Warehousing is an essential connection between producers and clients. Throughout the years warehousing has grown from a minor feature of a company s logistics framework to oneRead MoreBusiness Intelligence Systems1513 Words   |  7 Pagesand Business Strategy Module 2 Case Assignment: Business Intelligence Systems 04 June 2010 Business intelligence: Definition Business Intelligence (BI) is defined by IBM as, â€Å"the discipline that combines services, applications and technologies to gather, manage and analyze data, transforming it into usable information to develop insight and understanding needed to make informed decisions.† (IBM.com, 2006) In its most basic form, BI is an umbrella principle that synergizes the core understandingRead MoreEssay about Data Warehousing2843 Words   |  12 Pages Data Warehouses In the past decade, we have witnessed a computer revolution that was unimaginable. Ten to fifteen years ago, this world never would have imagined what computers would have done for business. Furthermore, the Internet and the ability to conduct electronic commerce have changed the way we are as consumers. One of the upcoming concepts of the computer revolution in the past ten years has been that of Data Warehousing. In the following pages, we will examineRead MoreTraditional Legacy Management Reporting And Decision Support Systems1516 Words   |  7 Pagesconnectivity and enterprise integration, supply chain, customer relationship management and business intelligence applications, not only improve the efficiency and effectiveness of the company’s supply network, but also satisfies business strategies such as customer satisfaction, consumer retention and product mix. This report concentrates on the features and security issues of these applications and eventually come up with a recommendation that Almost 21 should apply. 2.1 Supply Chain Network and SupplyRead MoreData Mining1668 Words   |  7 PagesData Mining: What is Data Mining? Overview Generally, data mining (sometimes called data or knowledge discovery) is the process of analyzing data from different perspectives and summarizing it into useful information - information that can be used to increase revenue, cuts costs, or both. Data mining software is one of a number of analytical tools for analyzing data. It allows users to analyze data from many different dimensions or angles, categorize it, and summarize the relationships identifiedRead MoreThe Data Warehouse Toolkit By Ralph Kimball And Margy Ross1421 Words   |  6 PagesASSIGNMENT -1 The Data Warehouse Toolkit Summary: The text book I have chosen is â€Å"The Data Warehouse Toolkit† third edition, written by Ralph Kimball and Margy Ross. This book mainly involves on techniques to develop the business in real-time. As the authors had a lot of experience because of their work from 1980’s, they have seen both the growth and failures of the companies in the market. Chapters in this text book involves goals of data warehousing which include Data staging area, data presentationRead MoreData Warehousing And Cloud Computing1664 Words   |  7 PagesINTRODUCTION This paper clearly illustrates the concepts of Data warehousing and Cloud computing. It also discusses the benefits and disadvantages of implementing Data Warehouse in a Cloud. Both cloud computing and data warehousing are the latest trends in modern computing. DWH is an integrated software component of the cloud and it provides timely support and accurate response to complex queries with Online Analytical Processing (OLAP) and data mining tools. Cloud computing provides reasonable speed

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