1.
Censoring (statistics)
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In statistics, engineering, economics, and medical research, censoring is a condition in which the value of a measurement or observation is only partially known. For example, suppose a study is conducted to measure the impact of a drug on mortality rate, in such a study, it may be known that an individuals age at death is at least 75 years. Such a situation could occur if the individual withdrew from the study at age 75, censoring also occurs when a value occurs outside the range of a measuring instrument. For example, a bathroom scale might only measure up to 300 pounds, if a 350 lb individual is weighed using the scale, the observer would only know that the individuals weight is at least 300 pounds. The problem of censored data, in which the value of some variable is partially known, is related to the problem of missing data. Censoring should not be confused with the related idea truncation, with censoring, observations result either in knowing the exact value that applies, or in knowing that the value lies within an interval. With truncation, observations never result in values outside a given range, note that in statistics, truncation is not the same as rounding. Left censoring – a data point is below a certain value, interval censoring – a data point is somewhere on an interval between two values. Right censoring – a data point is above a certain value, type I censoring occurs if an experiment has a set number of subjects or items and stops the experiment at a predetermined time, at which point any subjects remaining are right-censored. Random censoring is when each subject has a time that is statistically independent of their failure time. The observed value is the minimum of the censoring and failure times, interval censoring can occur when observing a value requires follow-ups or inspections. Left and right censoring are special cases of interval censoring, with the beginning of the interval at zero or the end at infinity, respectively. Estimation methods for using left-censored data vary, and not all methods of estimation may be applicable to, or the most reliable, a common misconception with time interval data is to class as left censored intervals where the start time is unknown. In these cases we have a bound on the time interval. Special techniques may be used to handle censored data, Tests with specific failure times are coded as actual failures, censored data are coded for the type of censoring and the known interval or limit. Special software programs can conduct a maximum likelihood estimation for summary statistics, confidence intervals, Reliability testing often consists of conducting a test on an item to determine the time it takes for a failure to occur. Sometimes a failure is planned and expected but does not occur, operator error, equipment malfunction, test anomaly, the test result was not the desired time-to-failure but can be used as a time-to-termination. The use of censored data is unintentional but necessary, sometimes engineers plan a test program so that, after a certain time limit or number of failures, all other tests will be terminated

2.
Reliability engineering
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Reliability engineering is engineering that emphasizes dependability in the lifecycle management of a product. Dependability, or reliability, describes the ability of a system or component to function under stated conditions for a period of time. Reliability may also describe the ability to function at a moment or interval of time. Reliability engineering represents a sub-discipline within systems engineering, Testability, Maintainability and maintenance are often defined as a part of reliability engineering in Reliability Programs. Reliability plays a key role in the cost-effectiveness of systems, Reliability engineering deals with the estimation, prevention and management of high levels of lifetime engineering uncertainty and risks of failure. Although stochastic parameters define and affect reliability, according to some authors on reliability engineering. You cannot really find a cause by only looking at statistics. Reliability engineering relates closely to safety engineering and to safety, in that they use common methods for their analysis. Reliability engineering focuses on costs of failure caused by system downtime, cost of spares, repair equipment, personnel, Safety engineering normally emphasizes not cost, but preserving life and nature, and therefore deals only with particular dangerous system-failure modes. High reliability levels also result from good engineering and from attention to detail, the word reliability can be traced back to 1816, by poet Coleridge. Before World War II the name has been linked mostly to repeatability, a test was considered reliable if the same results would be obtained repeatedly. The development of reliability engineering was here on a path with quality. The modern use of the reliability was defined by the U. S. military in the 1940s, characterizing a product that would operate when expected. In World War II, many reliability issues were due to inherent unreliability of electronics, in 1945, M. A. Miner published the seminal paper titled Cumulative Damage in Fatigue in an ASME journal. The IEEE formed the Reliability Society in 1948, in 1950, on the military side, a group called the Advisory Group on the Reliability of Electronic Equipment, AGREE, was born. The famous military standard 781 was created at that time, around this period also the much-used military handbook 217 was published by RCA and was used for the prediction of failure rates of components. The emphasis on component reliability and empirical research alone slowly decreases, More pragmatic approaches, as used in the consumer industries, are being used. In the 1980s, televisions were made up of solid-state semiconductors

3.
Life insurance
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Depending on the contract, other events such as terminal illness or critical illness can also trigger payment. The policy holder typically pays a premium, either regularly or as one lump sum, other expenses can also be included in the benefits. Life policies are legal contracts and the terms of the contract describe the limitations of the insured events. Specific exclusions are often written into the contract to limit the liability of the insurer, common examples are claims relating to suicide, fraud, war, riot, and civil commotion. Life-based contracts tend to fall into two categories, Protection policies – designed to provide a benefit, typically a lump sum payment. A common form of a protection policy design is term insurance, investment policies – where the main objective is to facilitate the growth of capital by regular or single premiums. Common forms are whole life, universal life, and variable life policies, an early form of life insurance dates to Ancient Rome, burial clubs covered the cost of members funeral expenses and assisted survivors financially. The first company to offer insurance in modern times was the Amicable Society for a Perpetual Assurance Office, founded in London in 1706 by William Talbot. Each member made a payment per share on one to three shares with consideration to age of the members being twelve to fifty-five. At the end of the year a portion of the contribution was divided among the wives and children of deceased members. The Amicable Society started with 2000 members and he was unsuccessful in his attempts at procuring a charter from the government. His disciple, Edward Rowe Mores, was able to establish the Society for Equitable Assurances on Lives, Mores also gave the name actuary to the chief official - the earliest known reference to the position as a business concern. The first modern actuary was William Morgan, who served from 1775 to 1830, in 1776 the Society carried out the first actuarial valuation of liabilities and subsequently distributed the first reversionary bonus and interim bonus among its members. It also used regular valuations to balance competing interests, the Society sought to treat its members equitably and the Directors tried to ensure that policyholders received a fair return on their investments. Premiums were regulated according to age, and anybody could be admitted regardless of their state of health, the sale of life insurance in the U. S. began in the 1760s. Between 1787 and 1837 more than two dozen life insurance companies were started, but fewer than half a dozen survived. S. The person responsible for making payments for a policy is the policy owner, the owner and insured may or may not be the same person. For example, if Joe buys a policy on his own life, but if Jane, his wife, buys a policy on Joes life, she is the owner and he is the insured

4.
Bathtub curve
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The bathtub curve is widely used in reliability engineering. It describes a form of the hazard function which comprises three parts, The first part is a decreasing failure rate, known as early failures. The second part is a constant failure rate, known as random failures, the third part is an increasing failure rate, known as wear-out failures. The name is derived from the shape of a bathtub, steep sides. In the mid-life of a product—generally, once it reaches consumers—the failure rate is low, in the late life of the product, the failure rate increases, as age and wear take their toll on the product. Many consumer product life cycles strongly exhibit the bathtub curve, the term Military Specification is often used to describe systems in which the infant mortality section of the bathtub curve has been burned out or removed. This is done mainly for life-critical or system-critical applications as it reduces the possibility of the system failing early in its life. Manufacturers will do this at some cost generally by means similar to environmental stress screening, in reliability engineering, the cumulative distribution function corresponding to a bathtub curve may be analysed using a Weibull chart. Gompertz–Makeham law of mortality Klutke, G. Kiessler, P. C, a critical look at the bathtub curve

5.
Poisson point process
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In probability, statistics and related fields, a Poisson point process or Poisson process is a type of random mathematical object that consists of points randomly located on a mathematical space. The Poisson point process is defined on the real line. In this setting, it is used, for example, in queueing theory to model events, such as the arrival of customers at a store or phone calls at an exchange. In this setting, the process is used in mathematical models and in the related fields of spatial point processes, stochastic geometry, spatial statistics. On more abstract spaces, the Poisson point process serves as an object of study in its own right. This has inspired the proposal of other point processes, some of which are constructed with the Poisson point process, the process is named after French mathematician Siméon Denis Poisson despite Poisson never having studied the process. The process was discovered independently and repeatedly in different settings, including experiments on radioactive decay, telephone call arrivals and insurance mathematics. The point process depends on a mathematical object, which, depending on the context, may be a constant, a locally integrable function or, in more general settings. In the first case, the constant, known as the rate or intensity, is the density of the points in the Poisson process located in some region of space. The resulting point process is called a homogeneous or stationary Poisson point process, depending on the setting, the process has several equivalent definitions as well as definitions of varying generality owing to its many applications and characterizations. Consequently, the notation, terminology and level of mathematical rigour used to define and study the Poisson point process, despite its different forms and varying generality, the Poisson point process has two key properties. The Poisson point process is related to the Poisson distribution, which implies that the probability of a Poisson random variable N being equal to n is given by, P = Λ n n. E − Λ where n. denotes n factorial and Λ is the single Poisson parameter that is used to define the Poisson distribution. If a Poisson point process is defined on some underlying space and this property is known under several names such as complete randomness, complete independence, or independent scattering and is common to all Poisson point processes. In other words, there is a lack of interaction between different regions and the points in general, which motivates the Poisson process being called a purely or completely random process. For all the instances of the Poisson point process, the two key properties of the Poisson distribution and complete independence play an important role, if a Poisson point process has a constant parameter, say, λ, then it is called a homogeneous or stationary Poisson point process. The parameter, called rate or intensity, is related to the number of Poisson points existing in some bounded region. The homogeneous Poisson point process, when considered on the positive half-line, can be defined as a process, a type of stochastic process