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In probability theory and statistics, the exponential distribution or negative exponential distribution is the probability distribution of the distance between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant average rate; the distance parameter could be any meaningful mono-dimensional measure of the process, such as time ...
Cumulative distribution function for the exponential distribution, often used as the cumulative failure function ().. To accurately model failures over time, a cumulative failure distribution, () must be defined, which can be any cumulative distribution function (CDF) that gradually increases from to .
Once the MTBF of a system is known, and assuming a constant failure rate, the probability that any one particular system will be operational for a given duration can be inferred [1] from the reliability function of the exponential distribution, () =.
In probability theory, an exponentially modified Gaussian distribution (EMG, also known as exGaussian distribution) describes the sum of independent normal and exponential random variables. An exGaussian random variable Z may be expressed as Z = X + Y , where X and Y are independent, X is Gaussian with mean μ and variance σ 2 , and Y is ...
A chi-squared distribution with 2 degrees of freedom (k = 2) is an exponential distribution with a mean value of 2 (rate λ = 1/2 .) A Weibull distribution with shape parameter k = 1 and rate parameter β is an exponential distribution with rate parameter β. A beta distribution with shape parameters α = β = 1 is a continuous uniform ...
A discrete probability distribution is applicable to the scenarios where the set of possible outcomes is discrete (e.g. a coin toss, a roll of a die) and the probabilities are encoded by a discrete list of the probabilities of the outcomes; in this case the discrete probability distribution is known as probability mass function.
Arrivals occur at rate λ according to a Poisson process and move the process from state i to i + 1. Service times have an exponential distribution with rate parameter μ in the M/M/1 queue, where 1/μ is the mean service time. All arrival times and services times are (usually) assumed to be independent of one another. [2]
A phase-type distribution is a probability distribution constructed by a convolution or mixture of exponential distributions. [1] 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 occurs may itself be a stochastic process.