In probability theory and statistics, the exponential distribution is the probability distribution of the time between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant average rate. It is a particular case of the gamma distribution. It is the continuous analogue of the geometric distribution, and it has the key property of being memoryless. In addition to being used for the analysis of Poisson point processes it is found in various other contexts.
In probability theory and statistics, the Weibull distribution is a continuous probability distribution. It is named after Swedish mathematician Waloddi Weibull, who described it in detail in 1951, although it was first identified by Fréchet (1927) and first applied by Rosin & Rammler (1933) to describe a particle size distribution.
In probability theory, a compound Poisson distribution is the probability distribution of the sum of a number of independent identically-distributed random variables, where the number of terms to be added is itself a Poisson-distributed variable. In the simplest cases, the result can be either a continuous or a discrete distribution.
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.
In probability theory the hypoexponential distribution or the generalized Erlang distribution is a continuous distribution, that has found use in the same fields as the Erlang distribution, such as queueing theory, teletraffic engineering and more generally in stochastic processes. It is called the hypoexponetial distribution as it has a coefficient of variation less than one, compared to the hyper-exponential distribution which has coefficient of variation greater than one and the exponential distribution which has coefficient of variation of one.
A phase-type distribution is a probability distribution constructed by a convolution or mixture of exponential distributions. It results from a system of one or more inter-related Poisson processes occurring in sequence, or phases. The sequence in which each of the phases occur may itself be a stochastic process. The distribution can be represented by a random variable describing the time until absorption of a Markov process with one absorbing state. Each of the states of the Markov process represents one of the phases.
The Gompertz–Makeham law states that the human death rate is the sum of an age-dependent component, which increases exponentially with age and an age-independent component. In a protected environment where external causes of death are rare, the age-independent mortality component is often negligible. In this case the formula simplifies to a Gompertz law of mortality. In 1825, Benjamin Gompertz proposed an exponential increase in death rates with age.
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.
In probability theory, the inverse Gaussian distribution is a two-parameter family of continuous probability distributions with support on (0,∞).
In probability theory and statistics, the normal-gamma distribution is a bivariate four-parameter family of continuous probability distributions. It is the conjugate prior of a normal distribution with unknown mean and precision.
The Weibull modulus is a dimensionless parameter of the Weibull distribution which is used to describe variability in measured material strength of brittle materials.
The Fréchet distribution, also known as inverse Weibull distribution, is a special case of the generalized extreme value distribution. It has the cumulative distribution function
In probability theory and statistics, the skew normal distribution is a continuous probability distribution that generalises the normal distribution to allow for non-zero skewness.
In probability theory and statistics, the Poisson distribution, named after French mathematician Siméon Denis Poisson, is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space if these events occur with a known constant mean rate and independently of the time since the last event. The Poisson distribution can also be used for the number of events in other specified intervals such as distance, area or volume.
The q-exponential distribution is a probability distribution arising from the maximization of the Tsallis entropy under appropriate constraints, including constraining the domain to be positive. It is one example of a Tsallis distribution. The q-exponential is a generalization of the exponential distribution in the same way that Tsallis entropy is a generalization of standard Boltzmann–Gibbs entropy or Shannon entropy. The exponential distribution is recovered as
The Lomax distribution, conditionally also called the Pareto Type II distribution, is a heavy-tail probability distribution used in business, economics, actuarial science, queueing theory and Internet traffic modeling. It is named after K. S. Lomax. It is essentially a Pareto distribution that has been shifted so that its support begins at zero.
In statistics and probability theory, the nonparametric skew is a statistic occasionally used with random variables that take real values. It is a measure of the skewness of a random variable's distribution—that is, the distribution's tendency to "lean" to one side or the other of the mean. Its calculation does not require any knowledge of the form of the underlying distribution—hence the name nonparametric. It has some desirable properties: it is zero for any symmetric distribution; it is unaffected by a scale shift; and it reveals either left- or right-skewness equally well. In some statistical samples it has been shown to be less powerful than the usual measures of skewness in detecting departures of the population from normality.
In statistics, the q-Weibull distribution is a probability distribution that generalizes the Weibull distribution and the Lomax distribution. It is one example of a Tsallis distribution.
Response modeling methodology (RMM) is a general platform for modeling monotonic convex relationships. RMM had been initially developed as a series of extensions to the original inverse Box–Cox transformation: where y is a percentile of the modeled response, Y, z is the respective percentile of a normal variate and λ is the Box–Cox parameter. As λ goes to zero, the inverse Box–Cox transformation becomes: an exponential model. Therefore, the original inverse Box-Cox transformation contains a trio of models: linear (λ = 1), power and exponential (λ = 0). This implies that on estimating λ, using sample data, the final model is not determined in advance but rather as a result of estimating. In other words, data alone determine the final model.
In probability theory, statistics, and machine learning, the continuous Bernoulli distribution is a family of continuous probability distributions parameterized by a single shape parameter , defined on the unit interval , by:
