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De Moivre's Law
De Moivre's Law is a survival model applied in actuarial science, named for Abraham de Moivre. It is a simple law of mortality based on a linear survival function. Definition De Moivre's law has a single parameter \omega called the ''ultimate age''. Under de Moivre's law, a newborn has probability of surviving at least ''x'' years given by the survival function : S(x) = 1 - \frac, \qquad 0 \leq x < \omega. In ''(x)'' denotes a status or life that has survived to age ''x'', and ''T''(''x'') is the future lifetime of ''(x)'' (''T''(''x'') is a random variable). The that ''(x)'' survives to age ''x+t'' is ''Pr [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Survival Model
Survival analysis is a branch of statistics for analyzing the expected duration of time until one event occurs, such as death in biological organisms and failure in mechanical systems. This topic is called reliability theory or reliability analysis in engineering, duration analysis or duration modelling in economics, and event history analysis in sociology. Survival analysis attempts to answer certain questions, such as what is the proportion of a population which will survive past a certain time? Of those that survive, at what rate will they die or fail? Can multiple causes of death or failure be taken into account? How do particular circumstances or characteristics increase or decrease the probability of survival? To answer such questions, it is necessary to define "lifetime". In the case of biological survival, death is unambiguous, but for mechanical reliability, failure may not be well-defined, for there may well be mechanical systems in which failure is partial, a matter of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Force Of Mortality
In actuarial science, force of mortality represents the instantaneous rate of mortality at a certain age measured on an annualized basis. It is identical in concept to failure rate, also called hazard function, in reliability theory. Motivation and definition In a life table, we consider the probability of a person dying from age ''x'' to ''x'' + 1, called ''q''''x''. In the continuous case, we could also consider the conditional probability of a person who has attained age (''x'') dying between ages ''x'' and ''x'' + ''Δx'', which is :P_(\Delta x)=P(x [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Benjamin Gompertz
Benjamin Gompertz (5 March 1779 – 14 July 1865) was a British self-educated mathematician and actuary, who became a Fellow of the Royal Society. Gompertz is now best known for his Gompertz law of mortality, a demographic model published in 1825. He was the brother of the early animal rights activist and inventor Lewis Gompertz and the poet Isaac Gompertz. Life Of the German Jewish family of Gompertz of Emmerich, he was born in London, where his father and grandfather had been successful diamond merchants. Debarred, as a Jew, from a university education, he studied on his own from an early age, in the writings of Isaac Newton, Colin Maclaurin, and William Emerson. From 1798 he was a prominent contributor to the ''Gentleman's Mathematical Companion'', and for a period won the annual prizes in the magazine for the solutions of problems. Gompertz married Abigail Montefiore (1790–1871) in 1810; they had three children. In line with his father's wishes, he entered the Londo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Continuous Probability Distribution
In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon in terms of its sample space and the probabilities of events (subsets of the sample space). For instance, if is used to denote the outcome of a coin toss ("the experiment"), then the probability distribution of would take the value 0.5 (1 in 2 or 1/2) for , and 0.5 for (assuming that the coin is fair). Examples of random phenomena include the weather conditions at some future date, the height of a randomly selected person, the fraction of male students in a school, the results of a survey to be conducted, etc. Introduction A probability distribution is a mathematical description of the probabilities of events, subsets of the sample space. The sample space, often denoted by \Omega, is the set of all possible outcomes of a random phe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Life Table
In actuarial science and demography, a life table (also called a mortality table or actuarial table) is a table which shows, for each age, what the probability is that a person of that age will die before their next birthday ("probability of death"). In other words, it represents the survivorship of people from a certain population. They can also be explained as a long-term mathematical way to measure a population's longevity. Tables have been created by demographers including Graunt, Reed and Merrell, Keyfitz, and Greville. There are two types of life tables used in actuarial science. The period life table represents mortality rates during a specific time period of a certain population. A cohort life table, often referred to as a generation life table, is used to represent the overall mortality rates of a certain population's entire lifetime. They must have had to be born during the same specific time interval. A cohort life table is more frequently used because it is able to ma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Failure Rate
Failure rate is the frequency with which an engineered system or component fails, expressed in failures per unit of time. It is usually denoted by the Greek letter λ (lambda) and is often used in reliability engineering. The failure rate of a system usually depends on time, with the rate varying over the life cycle of the system. For example, an automobile's failure rate in its fifth year of service may be many times greater than its failure rate during its first year of service. One does not expect to replace an exhaust pipe, overhaul the brakes, or have major transmission problems in a new vehicle. In practice, the mean time between failures (MTBF, 1/λ) is often reported instead of the failure rate. This is valid and useful if the failure rate may be assumed constant – often used for complex units / systems, electronics – and is a general agreement in some reliability standards (Military and Aerospace). It does in this case ''only'' relate to the flat region of the ba ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hazard Rate
Survival analysis is a branch of statistics for analyzing the expected duration of time until one event occurs, such as death in biological organisms and failure in mechanical systems. This topic is called reliability theory or reliability analysis in engineering Engineering is the use of scientific method, scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encompasses a broad rang ..., duration analysis or duration modelling in economics, and event history analysis in sociology. Survival analysis attempts to answer certain questions, such as what is the proportion of a population which will survive past a certain time? Of those that survive, at what rate will they die or fail? Can multiple causes of death or failure be taken into account? How do particular circumstances or characteristics increase or decrease the probability of survival? To answer such q ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Actuarial Notation
Actuarial notation is a shorthand method to allow actuaries to record mathematical formulas that deal with interest rates and life tables. Traditional notation uses a halo system where symbols are placed as superscript or subscript before or after the main letter. Example notation using the halo system can be seen below. Various proposals have been made to adopt a linear system where all the notation would be on a single line without the use of superscripts or subscripts. Such a method would be useful for computing where representation of the halo system can be extremely difficult. However, a standard linear system has yet to emerge. Example notation Interest rates \,i is the annual effective interest rate, which is the "true" rate of interest over ''a year''. Thus if the annual interest rate is 12% then \,i = 0.12. \,i^ (pronounced "i ''upper'' m") is the nominal interest rate convertible m times a year, and is numerically equal to m times the effective rate of interest over ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Uniform Distribution (continuous)
In probability theory and statistics, the continuous uniform distribution or rectangular distribution is a family of symmetric probability distributions. The distribution describes an experiment where there is an arbitrary outcome that lies between certain bounds. The bounds are defined by the parameters, ''a'' and ''b'', which are the minimum and maximum values. The interval can either be closed (e.g. , b or open (e.g. (a, b)). Therefore, the distribution is often abbreviated ''U'' (''a'', ''b''), where U stands for uniform distribution. The difference between the bounds defines the interval length; all intervals of the same length on the distribution's support are equally probable. It is the maximum entropy probability distribution for a random variable ''X'' under no constraint other than that it is contained in the distribution's support. Definitions Probability density function The probability density function of the continuous uniform distribution is: : f(x)=\begin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Life Tables
In actuarial science and demography, a life table (also called a mortality table or actuarial table) is a table which shows, for each age, what the probability is that a person of that age will die before their next birthday ("probability of death"). In other words, it represents the survivorship of people from a certain population. They can also be explained as a long-term mathematical way to measure a population's longevity. Tables have been created by demographers including Graunt, Reed and Merrell, Keyfitz, and Greville. There are two types of life tables used in actuarial science. The period life table represents mortality rates during a specific time period of a certain population. A cohort life table, often referred to as a generation life table, is used to represent the overall mortality rates of a certain population's entire lifetime. They must have had to be born during the same specific time interval. A cohort life table is more frequently used because it is able to ma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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