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Continuous probability distributions
Published in Andrew Metcalfe, David Green, Tony Greenfield, Mahayaudin Mansor, Andrew Smith, Jonathan Tuke, Statistics in Engineering, 2019
Andrew Metcalfe, David Green, Tony Greenfield, Mahayaudin Mansor, Andrew Smith, Jonathan Tuke
The Laplace distribution can be described as back-to-back exponential distributions. A random variable X with mean µ has the Laplace distribution with pdf f(x)=λ2e−λ|x−μ|,∞≤x≤∞.Sketch the pdf. Does the distribution have a mode?Calculate the variance and standard deviation.Calculate the probability that X is more than 1, 2, and 3 standard deviations away from the mean.Calculate the ratio of the inter-quartile range to the standard deviation and compare this with the value for a normal distribution.Explain why the difference of independent identically distributed exponential random variables has a Laplace distribution with mean 0.Explain how you can generate pseudo-random variates from a Laplace distribution given algorithms for generating pseudo-random variates from exponential and binomial distributions. Implement your method, construct a histogram, and check your answer to (e) when µ = 0 and λ = 2.Use the result of (e), and a function for generating pseudo-random exponential variates, to generate a million pseudo-random variates from a Laplace distribution .with mu = 0 and λ = 2. Construct a histogram, and check your answer to (e) when µ = 0 and λ = 2.
An estimation method for an unknown covariance in cross-section adjustment based on unbiased and consistent estimator
Published in Journal of Nuclear Science and Technology, 2023
Shuhei Maruyama, Tomohiro Endo, Akio Yamamoto
This probability distribution is a multivariate extension of the Laplace distribution. The Laplace distribution has a heavier tail than the normal distribution and is used to analyze data with outliers. The samples of generated based on this probability distribution show a mean of and a covariance of , as in the previous example. The means of samples for evaluated by the above procedure are shown in Table 3. The frequency distribution of is described in Figure 3 together with PDF. Again, we observed that the frequency of close to 1 increases as increases, but we found that the frequency distribution of no longer follows the distribution. However, the condition of Equations (24) and (25) are always valid even when a probability distribution other than the normal distribution is assumed.
Cauchy kernel-based maximum correntropy Kalman filter
Published in International Journal of Systems Science, 2020
Jiongqi Wang, Donghui Lyu, Zhangming He, Haiyin Zhou, Dayi Wang
.. It should be pointed out that the sign + means that the noises are drawn from with a probability of 0.8 and from with 0.2. obeys a Laplace distribution with the mean of 0 and variance of 1. i.e. . obeys the standard Cauchy distribution, i.e. .
On Double Lindley Distribution and Some of its Properties
Published in American Journal of Mathematical and Management Sciences, 2019
The Laplace distribution and its related versions have received much attention in the recent statistical literature. For example, see Dixit and Khandeparkar (2017, 2018) and Dixit and Subramanian (2016). The Laplace distribution is a two-tailed version of the exponential distribution and, hence, it is known in the literature as the double exponential distribution. Analogous to this, throughout here we consider a two-tailed version of Lindley distribution for getting a more flexible class of distributions, which we call the double Lindley distribution. We investigate several interesting properties of this distribution and highlight the relevance of such a class of distributions, compared to the Laplace distribution, by considering certain real life data sets.