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Probability Theory
Published in Paul L. Goethals, Natalie M. Scala, Daniel T. Bennett, Mathematics in Cyber Research, 2022
The Exponential distribution is a special case of the Gamma distribution. A random variable X is said to have a Gamma(α,λ) distribution if fX(x)=λαΓ(α)xα−1e−λx
Elements of Component Reliability
Published in M. Modarres, What Every Engineer Should Know About Reliability and Risk Analysis, 2018
The gamma distribution is obtained from the distribution of time to the occurrence of the kth event in a Poisson process. The gamma distribution is also suitable for representing the so-called shock models discussed earlier. In these models, an item fails after it receives a given number of shocks. If one assumes that the shocks occur according to Poisson process, a gamma distribution adequately models the distribution of time to failure of the item, given that it fails after receiving k shocks. According to our discussion in Section 3.2.1, the exponential distribution could also reasonably represent cases for which an item fails after receiving only one shock. Therefore, it is reasonable to expect exponential distribution to be a special case of gamma distribution.
The atmospheric subsystem
Published in Stephen A. Thompson, Hydrology for Water Management, 2017
Total annual precipitation tends to follow a normal distribution, particularly in humid climates. In arid and semiarid climates the proximity of zero (as the left boundary) produces a positively-skewed distribution. This raises the question of whether there might be a more robust theoretical distribution capable of handling both normal and positively-skewed data? One such distribution is the two-parameter gamma distribution (Table 3.4). The National Weather Service uses the gamma distribution to estimate annual precipitation probabilities. It is bounded by zero on the left and is unbounded on the right. The gamma distribution is defined by two parameters: a shape parameter (λ) and a scale parameter (β). The gamma PDF can vary from being J-shaped, to bell-shaped, to positively-skewed depending upon the values of the parameters. This makes the gamma more versatile than the normal distribution for modeling data. As with the normal distribution, the gamma can be evaluated by the use of tables. The EXCEL program has a wide variety of theoretical PDF and CDF distributions available as built-in functions including the normal and gamma. This is an extremely easy way to evaluate a distribution.
Statistical design of phase II exponential chart with estimated parameters under the unconditional and conditional perspectives using exact distribution of median run length
Published in Quality Technology & Quantitative Management, 2022
Because , the sum of independent and identically distributed exponential variables with parameter , follows a gamma distribution with the shape parameter and the scale parameter . Consequently, the random variable (r.v.) follows a chi-square distribution with degrees of freedom. Therefore, using Equation (4) and , we can write the (also called the conditional probability of a signal (CPS)) as a function of and as follows.
Total productive maintenance of make-to-stock production-inventory systems via artificial-intelligence-based iSMART
Published in International Journal of Systems Science: Operations & Logistics, 2021
Angelo Encapera, Abhijit Gosavi, Susan L. Murray
Results of using iSMART and RSMART on the TPM problem are presented here. The gamma distribution, as discussed above, is used for the time between failures, production time, and repair times, while the exponential distribution is used for the time between customer (demand) arrivals and the uniform distribution is used for the maintenance time. The mean of the gamma distribution parametrised by (n, λ) is given by n/λ, and its variance by n/λ2. The uniform distribution for the maintenance time will be characterised by (a, b), where a is the lower limit and b the upper limit. The exponential distribution for the time between arrivals will be defined by its single parameter μ, such that the mean is 1/μ. The inventory limit is defined by the upper and lower bounds (L, U).
Predictive maintenance scheduling with reliability characteristics depending on the phase of the machine life cycle
Published in Engineering Optimization, 2021
Iwona Paprocka, Wojciech M. Kempa, Bożena Skołud
In the method presented in this article, the failure-free time depends on the phase of the machine life cycle. The curve of the machine life cycle can be modelled by a traditional bathtub curve. In this case, phases I, II and III of the life cycle are modelled using the Weibull distribution (Kempa, Wosik, and Skołud 2011). However, the damage intensity function may also take other forms. In phase II, the frequency of failure is constant; thus, this phase can be modelled using the exponential distribution (Paprocka 2019). Phase III can be described by a log-normal distribution. Moreover, the gamma distribution is a flexible life distribution model which may offer a good fit to some sets of failure data. The gamma distribution arises naturally as the time-to-first-failure distribution for a system. The gamma distribution is also a good fit for the sum of independent exponential random variables (Kempa et al.2013). Considering the wide variety of probability distributions, it is necessary to search the distribution that best describes the historical data. In this article, the Maxwell distribution is used to describe failure-free operation time owing to its probability density function (PDF) having some geometric similarities to the gamma family of random variables (e.g. exponential, Erlang or chi-squared distributions). The Maxwell distribution is considered a lifetime distribution (e.g. Tyagi and Battacharya 1989a, 1989b), where the Bayesian and minimum variance unbiased estimators of its parameter are obtained. The case of Maxwell failure-free time distribution is investigated in Hossain and Huerta (2016), where, for example, the highest posterior density intervals and maximum likelihood estimators are found. Another contribution of this article is the application of the sample moments approach and renewal-theory approach for the Maxwell parameter estimation.