this paper presents a probabilistic optimal power flow (popf) algorithm considering different uncertainties in a smart grid. different uncertainties such as variation of nodal load, change in system configuration, measuring errors, forecasting errors, and etc. can be considered in the proposed algorithm. by increasing the penetration of the renewable energies in power systems, it is more essent...

one of the conventional methods for temporary support of tunnels is to use steel sets with shotcrete. the nature of a temporary support system demands a quick installation of its structures. as a result, the spacing between steel sets is not a fixed amount and it can be considered as a random variable. hence, in the reliability analysis of these types of structures, the selection of an appropri...

We present a relatively simple and mostly elementary proof of the L\'evy--Khintchine formula for subordinators. The main idea is to study Poisson process time-changed by subordinator. technical tools used are conditional expectations, probability generating function convergence discrete random variables.

Abstract: In this paper we consider the Gerber-Shiu penalty function in the compound binomial risk model with time-correlated claims. It is assumed that each main claim will induce a by-claim but the occurrence of the by-claim may be delayed with a certain probability. Formulas for the probability generating function of the penalty function are obtained, together with the expression for the pen...

The principle of the procedure to go from (4) to (5) is explained in the paper. The function S(z) given by (4) is a probability generating function (of the system contents of an output buffer during an arbitrary slot). Probability generating functions (pgf’s) have the property that they are analytic inside the unit disc, i.e. for {z : |z|6 1}. Among other things, this means that a pgf cannot ha...

Abstract: We explore conditions for a class of functions defined via an integral representation to be a probability generating function of some positive integer valued random variable. Interest in and research on this question is motivated by an apparently surprising connection between a family of classic shock models due to Esary et. al. (1973) and the negatively aging nonparametric notion of ...

Exact formulas for the mean and variance of the proportion of different types in a fixed generation of a multi-type Galton-Watson process are derived. The formulas are given in terms of iterates of the probability generating function of the offspring distribution. It is also shown that the sequence of types backwards from a randomly sampled particle in a fixed generation is a non-homogeneous Ma...

Example 5.5: Near-critical binary splitting. The theorem applies to binary splitting where the probability of division is p(z) = 1/2+1/(2z), and q(z) = 1− p(z) is the probability of no children, if population size is z. Thus, m(z) = 1 + 1/z and the process is supercritical, but approaches criticality as z increases. The probability generating function of the offspring number is fz(s) = q(z) + p...

This paper calculates probability distributions modeling the Luria-Delbrück experiment. We show that by thinking purely in terms of generating functions, and using a ‘backwards in time’ paradigm, that formulas describing various situations can be easily obtained. This includes a generating function for Haldane’s probability distribution due to Ycart. We apply our formulas to both simulated and ...