###### Graphics processing units in acceleration of bandwidth selection for kernel density estimation

Workshop on General Purpose Processing on Graphics Processing Units, GPGPU-4, Newport Beach, CA, USA, pp. 9:1-9:9. Gramacki, A., Gramacki, J. and Andrzejewski, W. (2010). Probability density functions for calculating approximate aggregates, Foundations of Computing and Decision Sciences 35(4): 223-240. Greengard, L. and Strain, J. (1991). The fast Gauss transform, SIAM Journal on Scientific and Statistical Computing 12(1): 79-94. Harris, M. (2007). Optimizing parallel reduction in CUDA, http

###### Computing the two first probability density functions of the random Cauchy-Euler differential equation: Study about regular-singular points

_W(u)=\mathbb{E}\left[W(u)\right] \end{array}$ , and its variance, σ W 2 ( u ) = V [ W ( u ) ] $\begin{array}{} \displaystyle \sigma_W^2(u)=\mathbb{V}\left[W(u)\right] \end{array}$ . Additionally, the computation of the first probability density function (1-PDF), f ^ 1 ( w , u ) $\begin{array}{} \displaystyle \hat{f}_1(w,u) \end{array}$ , is also important because from it one gets a full probabilistic description of the solution SP in each time instant u . Furthermore, from the 1-PDF one can compute all the one-dimensional statistical moments of W ( u ), E [ ( W ( u ) ) k ] = ∫ − ∞ ∞ w

###### Bayesian Analysis of a Simple Measurement Model Distinguishing between Types of Information

comparison. Metrologia 46, 261-266. [7] Lira, I., Grientschnig, D. (2013). A formalism for expressing the probability density functions of interrelated quantities. Measurement Science Review 13, 50-55. [8] Sun, D., Berger, J. O. (1998). Reference priors with partial information. Biometrika 85, 55-71. [9] Lira, I., Grientschnig, D. (2010). Equivalence of alternative Bayesian procedures for evaluating measurement uncertainty. Metrologia 47, 334-336. [10] Possolo, A., Toman, B. (2007). Assessment of measurement

###### Analysis of the Distribution Influence of the Density of Cost-forming Factors on Results of the LCCA Calculations

## Abstract

The paper evaluates the relationship between the selection of the probability density function and the construction price, and the price of the building's life cycle, in relation to the deterministic cost estimate in terms of the minimum, mean, and maximum. The deterministic cost estimates were made based on the minimum, mean, and maximum prices: labor rates, indirect costs, profit, and the cost of equipment and materials. The net construction prices received were given different probability density distributions based on the minimum, mean, and maximum values. Twelve kinds of probability distributions were used: triangular, normal, lognormal, beta pert, gamma, beta, exponential, Laplace, Cauchy, Gumbel, Rayleigh, and uniform. The results of calculations with the event probability from 5 to 95% were subjected to the statistical comparative analysis. The dependencies between the results of calculations were determined, for which different probability density distributions of price factors were assumed. A certain price level was assigned to specific distributions in 6 groups based on the t-test. It was shown that each of the distributions analyzed is suitable for use, however, it has consequences in the form of a final result. The lowest final price is obtained using the gamma distribution, the highest is obtained by the beta distribution, beta pert, normal, and uniform.

###### Probability Density Functions of Voltage Sags Measured Indices

## Probability Density Functions of Voltage Sags Measured Indices

Voltage sags can cause interruptions of industrial processes, which could result as a malfunction of equipment and considerable economic losses. Thus, it is very useful to see certain rules of voltage sags occurrence due to duration and depth.

This paper presents statistical analyses of voltage sags in several domestic and industrial transformer stations. Voltage sag probability functions are calculated from actual measurement data, by means of a hill climbing algorithm. Lognormal and Weibull frequency distribution functions are used to describe distribution of measured voltage dips.

###### A Simple Discrete Approximation for the Renewal Function

## Abstract

**Background: **The renewal function is widely useful in the areas of reliability, maintenance and spare component inventory planning. Its calculation relies on the type of the probability density function of component failure times which can be, regarding the region of the component lifetime, modelled either by the exponential or by one of the peak-shaped density functions. For most peak-shaped distribution families the closed form of the renewal function is not available. Many approximate solutions can be found in the literature, but calculations are often tedious. Simple formulas are usually obtained for a limited range of functions only. **Objectives: **We propose a new approach for evaluation of the renewal function by the use of a simple discrete approximation method, applicable to any probability density function. **Methods/Approach: **The approximation is based on the well known renewal equation. **Results: **The usefulness is proved through some numerical results using the normal, lognormal, Weibull and gamma density functions. The accuracy is analysed using the normal density function. **Conclusions: **The approximation proposed enables simple and fairly accurate calculation of the renewal function irrespective of the type of the probability density function. It is especially applicable to the peak-shaped density functions when the analytical solution hardly ever exists.

###### A Formalism for Expressing the Probability Density Functions of Interrelated Quantities

In this paper we address measurement problems involving several quantities that are interrelated by model equations. Available knowledge about some of these quantities is represented by probability density functions (PDFs), which are then propagated through the model in order to obtain the PDFs attributed to the quantities for which nothing is initially known. A formalism for analyzing such models is presented. It comprises the concept of a „base parameterization“, which is used in conjunction with the change-of-variables theorem. The calculation procedure that results from this formalism is described in very general terms. Guidance is given on how to employ it in practice by presenting both an elementary example and a much more involved one.

###### Dynamic Multi-Attribute Group Decision Making Model Based on Generalized Interval-Valued Trapezoidal Fuzzy Numbers

## Abstract

In this paper we investigate the dynamic multi-attribute group decision making problems, in which all the attribute values are provided by multiple decision makers at different periods. In order to increase the level of overall satisfaction for the final decision and deal with uncertainty, the attribute values are enhanced with generalized interval-valued trapezoidal fuzzy numbers to cope with the vagueness and indeterminacy. We first define the Dynamic Generalized Interval-valued Trapezoidal Fuzzy Numbers Weighted Geometric Aggregation (DGITFNWGA) operator and give an approach to determine the weights of periods, using the probability density function of Gamma distribution, and then a dynamic multi-attribute group decision making method is developed. The method proposed employs the Generalized Interval-valued Trapezoidal Fuzzy Numbers Hybrid Geometric Aggregation (GITFNHGA) operator to aggregate all individual decision information into the collective attribute values corresponding to each alternative at the same time period, and then utilizes the DGITFNWGA operator to aggregate the collective attribute values at different periods into the overall attribute values corresponding to each alternative and obtains the alternatives ranking, by which the optimal alternative can be determined. Finally, an illustrative example is given to verify the approach developed.

######
On The Distribution Of Mixed Sum Of Independent Random Variables One Of Them Associated With Srivastava's Polynomials And *H* -Function

## Abstract

In this paper, we obtain the distribution of mixed sum of two independent random variables with different probability density functions. One with probability density function defined in finite range and the other with probability density function defined in infinite range and associated with product of Srivastava's polynomials and H-function. We use the Laplace transform and its inverse to obtain our main result. The result obtained here is quite general in nature and is capable of yielding a large number of corresponding new and known results merely by specializing the parameters involved therein. To illustrate, some special cases of our main result are also given.

###### Analysis of Reliability and Stability of Bar Structures / Analiza Niezawodnosci I Statecznosci Konstrukcji Pretowej

## Abstract

In the paper, the Hasofer-Lind index is applied for determining the probability of stability loss of truss structure under random load. In 1974 Hasofer-Lind proposed a modified reliability index that did not exhibit the invariance problem. The “correction” is the evaluation the limit state function at a point known as the “design point”, instead of the mean values. The design point is generally not known a priori, an iteration technique must be used to find out the reliability index. The paper shows how the reliability index changes under the influence of different variables mean value, standard deviation, and probability density function