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Table Of Integrals Series And Products

Tables of integrals

(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

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An Introduction To The Theory Of Point Processes

Calderas are collapse structures related to the emptying of magmatic reservoirs, often associated with large eruptions from long-lived magmatic systems. Understanding how magma is transferred from a magma reservoir to the surface before eruptions is a major challenge. Here we exploit the historical, archaeological and geological record of Campi Flegrei caldera to estimate the surface deformation preceding the Monte Nuovo eruption and investigate the shallow magma transfer. Our data suggest a progressive magma accumulation from ~1251 to 1536 in a 4.6 ± 0.9 km deep source below the caldera centre, and its transfer, between 1536 and 1538, to a 3.8 ± 0.6 km deep magmatic source ~4 km NW of the caldera centre, below Monte Nuovo; this peripheral source fed the eruption through a shallower source, 0.4 ± 0.3 km deep. This is the first reconstruction of pre-eruptive magma transfer at Campi Flegrei and corroborates the existence of a stationary oblate source, below the caldera centre, that has been feeding lateral eruptions for the last ~5 ka. Our results suggest: 1) repeated emplacement of magma through intrusions below the caldera centre; 2) occasional lateral transfer of magma feeding non-central eruptions within the caldera. Comparison with historical unrest at calderas worldwide suggests that this behavior is common.

Magma transfer at Campi Flegrei caldera (Italy) before the 1538 AD eruption

For nonnegative random variables with ﬁnite means we introduce an analogous of theequilibrium residual-lifetime distribution based on the quantile function. This allows toconstruct new distributions with support (0,1), and to obtain a new quantile-based versionof the probabilistic generalization of Taylor’s theorem. Similarly, for pairs of stochasticallyordered random variables we come to a new quantile-based form of the probabilistic meanvalue theorem. The latter involves a distribution that generalizes the Lorenz curve. Weinvestigate the special case of proportional quantile functions and apply the given resultsto various models based on classes of distributions and measures of risk theory. Motivatedby some stochastic comparisons, we also introduce the ‘expected reversed proportionalshortfall order’, and a new characterization of random lifetimes involving the reversedhazard rate function.Short title: A quantile-based probabilistic mean value theorem. 1 Introduction The quantile function, being the inverse of the cumulative distribution function of a randomvariable, is often invoked in applied probability and statistics. In certain cases the approachbased on quantile functions is more fruitful than the use of cumulative distribution functions,since quantile functions are less inﬂuenced by extreme statistical observations. For instance,quantile functions can be properly employed to formulate properties of entropy function and

/pdf/a-quantile-based-probabilistic-mean-value-theorem-3vbsf8x89s.pdf

A quantile-based probabilistic mean value theorem

In this paper we study the distribution of the location, at time t , of a particle moving U time units upwards, V time units downwards, and W time units of no movement (idle). These are repeated cyclically, according to independent alternating renewals. The distributions of U , V , and W are absolutely continuous. The velocities are v = +1 upwards, v = -1 downwards, and v = 0 during idle periods. Let Y + ( t ), Y − ( t ), and Y 0 ( t ) denote the total time in (0, t ) of movements upwards, downwards, and no movements, respectively. The exact distribution of Y + ( t ) is derived. We also obtain the probability law of X ( t ) = Y + ( t ) - Y − ( t ), which describes the particle's location at time t . Explicit formulae are derived for the cases of exponential distributions with equal rates, with different rates, and with linear rates (leading to damped processes).

/pdf/generalized-telegraph-process-with-random-delays-fagh2jiwfb.pdf

Generalized telegraph process with random delays

We consider a first-passage-time problem for a compound Poisson process characterized by independent, identically and exponentially distributed jumps, occurring according to the power-law process (PLP). First of all, we refer to the conditional product moments of arrival times and to the interarrival times density of a power-law process. We then obtain the probability density of the crossing time through a linear boundary at the occurrence of the nth jump. In particular, we express the first-passage-time density in terms of a conditional expectation involving the arrival times.

On a first-passage-time problem for the compound power-law process

We consider a bilateral birth-death process characterized by a constant transition rate λ from even states and a possibly different transition rate μ from odd states. We determine the probability generating functions of the even and odd states, the transition probabilities, mean and variance of the process for arbitrary initial state. Some features of the birth-death process confined to the non-negative integers by a reflecting boundary in the zero-state are also analyzed. In particular, making use of a Laplace transform approach we obtain a series form of the transition probability from state 1 to the zero-state.

On a bilateral birth-death process with alternating rates

We propose a stochastic model for the firing activity of a neuronal unit. It includes the decay effect of the membrane potential in absence of stimuli, and the occurrence of time-varying excitatory inputs governed by a Poisson process. The sample-paths of the membrane potential are piecewise exponentially decaying curves with jumps of random amplitudes occurring at the input times. An analysis of the probability distributions of the membrane potential and of the firing time is performed. In the special case of time-homogeneous stimuli the firing density is obtained in closed form, together with its mean and variance.

Barbara Martinucci

Papers

A quantile-based probabilistic mean value theorem

Generalized telegraph process with random delays

On a first-passage-time problem for the compound power-law process

On a bilateral birth-death process with alternating rates

Analysis of a stochastic neuronal model with excitatory inputs and state-dependent effects.