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Table Of Integrals Series And Products

Tables of integrals

The Theory of Stochastic Processes. By D. R. Cox and H. D. Miller. Pp. x, 398. 70s. (Methuen)

(1987). Applied Probability and Queues. Journal of the Operational Research Society: Vol. 38, No. 11, pp. 1095-1096.

https://link.springer.com/content/pdf/10.1057%2Fjors.1987.184.pdf

Applied Probability and Queues

Stochastic comparisons of series and parallel systems with randomized independent components

We consider a standard Brownian motion whose drift alternates randomly between a positive and a negative value, according to a generalized telegraph process. We first investigate the distribution of the occupation time, i.e. the fraction of time when the motion moves with positive drift. This allows to obtain explicitly the probability law and the flow function of the random motion. We discuss three special cases when the times separating consecutive drift changes have (i) exponential distribution with constant rates, (ii) Erlang distribution, and (iii) exponential distribution with linear rates. In conclusion, in view of an application in environmental sciences we evaluate the density of a Wiener process with infinitesimal moments alternating at inverse Gaussian distributed random times.

Probability Law and Flow Function of Brownian Motion Driven by a Generalized Telegraph Process

The cumulative entropy is an information measure which is alternative to the differential entropy and is connected with a notion in reliability theory. Indeed, the cumulative entropy of a random lifetime X can be expressed as the expectation of its mean inactivity time evaluated at X. After a brief review of its main properties, in this paper, we relate the cumulative entropy to the cumulative inaccuracy and provide some inequalities based on suitable stochastic orderings. We also show a characterization property of the dynamic version of the cumulative entropy. In conclusion, a stochastic comparison between the empirical cumulative entropy and the empirical cumulative inaccuracy is investigated.

Stochastic Comparisons of Cumulative Entropies

We consider dynamic versions of the mutual information of lifetime distributions, with a focus on past lifetimes, residual lifetimes, and mixed lifetimes evaluated at different instants. This allows us to study multicomponent systems, by measuring the dependence in conditional lifetimes of two components having possibly different ages. We provide some bounds, and investigate the mutual information of residual lifetimes within the time-transformed exponential model (under both the assumptions of unbounded and truncated lifetimes). Moreover, with reference to the order statistics of a random sample, we evaluate explicitly the mutual information between the minimum and the maximum, conditional on inspection at different times, and show that it is distribution-free in a special case. Finally, we develop a copula-based approach aiming to express the dynamic mutual information for past and residual bivariate lifetimes in an alternative way.

Antonio Di Crescenzo

Papers

Stochastic comparisons of series and parallel systems with randomized independent components

Probability Law and Flow Function of Brownian Motion Driven by a Generalized Telegraph Process

Stochastic Comparisons of Cumulative Entropies

On dynamic mutual information for bivariate lifetimes

M/M/1 queue in two alternating environments and its heavy traffic approximation