Probability Density
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Dr. Uppu Venkata Subbarao

Abstract: In this paper we investigated the order statistics by using Additive Uniform Exponential Distribution (AUED) proposed by Venkata Subbarao Uppu (2010).The probability density functions of rth order Statistics, lth moment of the rth order Statistic, minimum, maximum order statistics, mean of the maximum and minimum order statistics, the joint density function of two order statistics were calculated and discussed in detailed . Applications and several aspects were discussed Keywords: Additive Uniform Exponential Distribution, Moments, Minimum order statistic, Maximum order statistic, Joint density of the order Statistics, complete length of service.


In this paper, we introduce a dynamical model for the time evolution of probability density functions incorporating uncertainty in the parameters. The uncertainty follows stochastic processes, thereby defining a new class of stochastic processes with values in the space of probability densities. The purpose is to quantify uncertainty that can be used for probabilistic forecasting. Starting from a set of traded prices of equity indices, we do some empirical studies. We apply our dynamic probabilistic forecasting to option pricing, where our proposed notion of model uncertainty reduces to uncertainty on future volatility. A distribution of option prices follows, reflecting the uncertainty on the distribution of the underlying prices. We associate measures of model uncertainty of prices in the sense of Cont.

Maria V Clavijo ◽  
Adriana M Schleder ◽  
Enrique Lopez Droguett ◽  
Marcelo R Martins

Currently, a Dynamic Position (DP) System is commonly used for offshore operations. However, DP failures may generate environmental and economic losses; thus, this paper presents the Reliability, Availability and Maintainability (RAM) analysis for two different generations of DP system (DP2 and DP3) used in drilling operations. In addition to the RAM analysis, the approach proposed herein considers the uncertainties present in the equipment failure data and provides more information about criticality equipment ratings and probability density functions (pdf) of the repair times. The reliability analysis shows that, for 3 months of operation, the total failure probability of the DP2 system is 1.52% whereas this probability for the DP3 system is only 0.16%. The results reveal that the bus-bar is the most critical equipment of the DP2 system, whereas the wind sensor represents the priority equipment in the DP3 system. Using 90% confidence level, each DP configuration was evaluated for a 1-year operation, finding a reliability mean equal to 70.39% and 86.77% for the DP2 system and the DP3 system, respectively. The DP2 system asymptotic availability tends to present a constant value of 99.98% whereas for the DP3 system, it tends to be 99.99%. Finally, the maintainability analysis allows concluding that the mean time for system repair is expected to be 3.6 h. This paper presents a logical pathway for analysts, operators, and reliability engineers of the oil and gas industry.

2021 ◽  
Vol 11 (20) ◽  
pp. 9695
Jun Lei ◽  
José Antonio Lozano-Galant ◽  
Dong Xu ◽  
Feng-Liang Zhang ◽  
Jose Turmo

Deflections are commonly measured in the static structural system identification of structures. Comparatively less attention has been paid to the possibility of measuring rotations for structural system identification purposes, despite the many advantages of using inclinometers, such as a high resolution and being reference free. Although some work using rotations can be found in the literature, this paper, for the very first time, proposes a statistical analysis that justifies the theoretical advantage of measuring rotations. The analytical expressions for the target parameters are obtained via static structural system identification using the constrained observability method first. Combined with the inverse distribution theory, the probability density function of the estimations of the target parameters can be obtained. Comparative studies on a simply supported bridge and a frame structure demonstrate the advantage of measuring rotations regarding the unbiasedness and the extent of variation in the estimations. To achieve robust parameter estimations, four strategies to use redundant rotations are proposed and compared. Numerical verifications on a bridge structure and a high-rise building have shown promising results.

2021 ◽  
Tim C Jenkins

Abstract Superposed wavefunctions in quantum mechanics lead to a squared amplitude that introduces interference into a probability density, which has long been a puzzle because interference between probability densities exists nowhere else in probability theory. In recent years, Man’ko and coauthors have successfully reconciled quantum and classical probability using a symplectic tomographic model. Nevertheless, there remains an unexplained coincidence in quantum mechanics, namely, that mathematically, the interference term in the squared amplitude of superposed wavefunctions has the form of a variance of a sum of correlated random variables, and we examine whether there could be an archetypical variable behind quantum probability that provides a mathematical foundation that observes both quantum and classical probability directly. The properties that would need to be satisfied for this to be the case are identified, and a generic variable that satisfies them is found that would be present everywhere, transforming into a process-specific variable wherever a quantum process is active. This hidden generic variable appears to be such an archetype.

2021 ◽  
Vol 31 (6) ◽  
Nadhir Ben Rached ◽  
Abdul-Lateef Haji-Ali ◽  
Gerardo Rubino ◽  
Raúl Tempone

AbstractWe discuss estimating the probability that the sum of nonnegative independent and identically distributed random variables falls below a given threshold, i.e., $$\mathbb {P}(\sum _{i=1}^{N}{X_i} \le \gamma )$$ P ( ∑ i = 1 N X i ≤ γ ) , via importance sampling (IS). We are particularly interested in the rare event regime when N is large and/or $$\gamma $$ γ is small. The exponential twisting is a popular technique for similar problems that, in most cases, compares favorably to other estimators. However, it has some limitations: (i) It assumes the knowledge of the moment-generating function of $$X_i$$ X i and (ii) sampling under the new IS PDF is not straightforward and might be expensive. The aim of this work is to propose an alternative IS PDF that approximately yields, for certain classes of distributions and in the rare event regime, at least the same performance as the exponential twisting technique and, at the same time, does not introduce serious limitations. The first class includes distributions whose probability density functions (PDFs) are asymptotically equivalent, as $$x \rightarrow 0$$ x → 0 , to $$bx^{p}$$ b x p , for $$p>-1$$ p > - 1 and $$b>0$$ b > 0 . For this class of distributions, the Gamma IS PDF with appropriately chosen parameters retrieves approximately, in the rare event regime corresponding to small values of $$\gamma $$ γ and/or large values of N, the same performance of the estimator based on the use of the exponential twisting technique. In the second class, we consider the Log-normal setting, whose PDF at zero vanishes faster than any polynomial, and we show numerically that a Gamma IS PDF with optimized parameters clearly outperforms the exponential twisting IS PDF. Numerical experiments validate the efficiency of the proposed estimator in delivering a highly accurate estimate in the regime of large N and/or small $$\gamma $$ γ .

Iwok Iberedem Aniefiok ◽  
Barinaadaa John Nwikpe

In this paper, a new continuous probability distribution named Iwok-Nwikpe distribution is proposed. Some essential statistical properties of the proposed probability distribution have been derived. The graphs of the survival function, probability density function (p.d.f) and cumulative distribution function (c.d.f) were plotted at different values of the parameter. The mathematical expression for the moment generating function (mgf) was derived. Consequently, the first three crude moments were obtained; the distribution of order statistics, the second and third moments corrected for the mean have also been derived. The parameter of the Iwok-Nwikpe distribution was estimated by means of maximum likelihood technique. To establish the goodness of fit of the Iwok-Nwikpe distribution, three real data sets from engineering and medical science were fitted to the distribution. Findings of the study revealed that the Iwok-Nwikpe distribution performed better than the one parameter exponential distribution and other competing models used for the study.

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