The encounter probability for mountain slope hazards

1999 ◽  
Vol 36 (6) ◽  
pp. 1195-1196 ◽  
Author(s):  
D M McClung
Keyword(s):  
Poisson Process ◽  
Return Period ◽  
Finite Time ◽  
Bernoulli Distribution ◽  
Mountain Slope ◽  
Encounter Probability ◽  
Time Period

In typical risk calculations for the mountain slope hazards one wishes to calculate the encounter probability: the probability of facilities or vehicles being hit at least once when exposed for a finite time period L with events having a return period T at a location. In this note, it is assumed that the events are rare, independent, and discrete, with arrival according to a binomial (or Bernoulli) distribution or a Poisson process. The constraints on the formulations for the processes are provided and it is shown that for typical applications either assumption (binomial or Poisson process) may be used in practice almost interchangeably.

scholarly journals The fighter problem: optimal allocation of a discrete commodity

2011 ◽  
Vol 43 (01) ◽  
pp. 121-130 ◽  
Author(s):  
Jay Bartroff ◽  
Ester Samuel-Cahn
Keyword(s):  
Poisson Process ◽  
Optimal Allocation ◽  
Optimal Number ◽  
Homogeneous Poisson Process ◽  
Time Period

In this paper we study the fighter problem with discrete ammunition. An aircraft (fighter) equipped with n anti-aircraft missiles is intercepted by enemy airplanes, the appearance of which follows a homogeneous Poisson process with known intensity. If j of the n missiles are spent at an encounter, they destroy an enemy plane with probability a(j), where a(0) = 0 and {a(j)} is a known, strictly increasing concave sequence, e.g. a(j) = 1 - q j , 0 < q < 1. If the enemy is not destroyed, the enemy shoots the fighter down with known probability 1 - u, where 0 ≤ u ≤ 1. The goal of the fighter is to shoot down as many enemy airplanes as possible during a given time period [0, T]. Let K(n, t) be the smallest optimal number of missiles to be used at a present encounter, when the fighter has flying time t remaining and n missiles remaining. Three seemingly obvious properties of K(n, t) have been conjectured: (A) the closer to the destination, the more of the n missiles one should use; (B) the more missiles one has; the more one should use; and (C) the more missiles one has, the more one should save for possible future encounters. We show that (C) holds for all 0 ≤ u ≤ 1, that (A) and (B) hold for the ‘invincible fighter’ (u = 1), and that (A) holds but (B) fails for the ‘frail fighter’ (u = 0); the latter is shown through a surprising counterexample, which is also valid for small u > 0 values.

scholarly journals The fighter problem: optimal allocation of a discrete commodity

2011 ◽  
Vol 43 (1) ◽  
pp. 121-130 ◽  
Author(s):  
Jay Bartroff ◽  
Ester Samuel-Cahn
Keyword(s):  
Poisson Process ◽  
Optimal Allocation ◽  
Optimal Number ◽  
Homogeneous Poisson Process ◽  
Time Period

In this paper we study the fighter problem with discrete ammunition. An aircraft (fighter) equipped with n anti-aircraft missiles is intercepted by enemy airplanes, the appearance of which follows a homogeneous Poisson process with known intensity. If j of the n missiles are spent at an encounter, they destroy an enemy plane with probability a(j), where a(0) = 0 and {a(j)} is a known, strictly increasing concave sequence, e.g. a(j) = 1 - qj, 0 < q < 1. If the enemy is not destroyed, the enemy shoots the fighter down with known probability 1 - u, where 0 ≤ u ≤ 1. The goal of the fighter is to shoot down as many enemy airplanes as possible during a given time period [0, T]. Let K(n, t) be the smallest optimal number of missiles to be used at a present encounter, when the fighter has flying time t remaining and n missiles remaining. Three seemingly obvious properties of K(n, t) have been conjectured: (A) the closer to the destination, the more of the n missiles one should use; (B) the more missiles one has; the more one should use; and (C) the more missiles one has, the more one should save for possible future encounters. We show that (C) holds for all 0 ≤ u ≤ 1, that (A) and (B) hold for the ‘invincible fighter’ (u = 1), and that (A) holds but (B) fails for the ‘frail fighter’ (u = 0); the latter is shown through a surprising counterexample, which is also valid for small u > 0 values.

Stochastic processes involving random deletion

2001 ◽  
Vol 38 (01) ◽  
pp. 95-107 ◽  
Author(s):  
Mark D. Rothmann ◽  
Hammou El Barmi
Keyword(s):  
Stochastic Processes ◽  
Poisson Process ◽  
Finite Time ◽  
Empirical Distribution ◽  
Nonhomogeneous Poisson Process

We consider a system where units having magnitudes arrive according to a nonhomogeneous Poisson process, remain there for a random period and then depart. Eventually, at any point in time only a portion of those units which have entered the system remain. Of interest are the finite time properties and limiting behaviors of the distribution of magnitudes among the units present in the system and among those which have departed from the system. We will derive limiting results for the empirical distribution of magnitudes among the active (departed) units. These results are also shown to extend to systems having stages or steps through which units must proceed. Examples are given to illustrate these results.

The Ruin Probability During a Finite Time Period

Risk Theory ◽  
1977 ◽  
pp. 132-136
Author(s):  
Robert Eric Beard ◽  
Teivo Pentikäinen ◽  
Erkki Pesonen
Keyword(s):  
Finite Time ◽  
Ruin Probability ◽  
Time Period

Adjusting Environmental Contours for Specified Expected Number of Unwanted Events

2020 ◽  
Author(s):  
Erik Vanem
Keyword(s):  
Reliability Analysis ◽  
Return Period ◽  
Environmental Conditions ◽  
Structural Reliability ◽  
Probability Of Failure ◽  
Expected Number ◽  
Probability Of Exceedance ◽  
Underlying Distribution ◽  
Extreme Loads ◽  
Time Period

Abstract Environmental contours are applied in probabilistic structural reliability analysis to identify extreme environmental conditions that may give rise to extreme loads and responses. Typically, they are constructed to correspond to a certain return period and a probability of exceedance with regards to the environmental conditions that can again be related to the probability of failure of a structure. Thus, they describe events with a certain probability of being exceeded one or more times during a certain time period, which can be found from a certain percentile of the underlying distribution. In this paper, various ways of adjusting such environmental contours to account for the expected number of exceedances within a certain time period are discussed. Depending on how such criteria are defined, one may get more lenient or more stringent criteria compared to the classical return period.

Probabilities of ruin when the safety loading tends to zero

2000 ◽  
Vol 32 (3) ◽  
pp. 885-923 ◽  
Author(s):  
Vsevolod K. Malinovskii
Keyword(s):  
Finite Time ◽  
Absolute Constant ◽  
Classical Result ◽  
Risk Theory ◽  
Premium Rate ◽  
Collective Risk ◽  
Time Period ◽  
Positive Absolute Constant ◽  
Uniform Approximations ◽  
Insurance Business

When the premium rate is a positive absolute constant throughout the time period of observation and the safety loading of the insurance business is positive, a classical result of collective risk theory claims that probabilities of ultimate ruin ψ(u) and of ruin within finite time ψ(t,u) decrease as eϰu with a constant ϰ>0, as the initial risk reserve u increases. This paper establishes uniform approximations to ψ(t,u) with slower rates of decrease when the premium rate depends on u in such a way that the safety loading decreases to zero as u→∞.

scholarly journals Flood frequency analysis supported by the largest historical flood

2014 ◽  
Vol 14 (6) ◽  
pp. 1543-1551 ◽  
Author(s):  
W. G. Strupczewski ◽  
K. Kochanek ◽  
E. Bogdanowicz
Keyword(s):  
Return Period ◽  
Historical Record ◽  
Flood Frequency ◽  
Flood Frequency Analysis ◽  
Historical Period ◽  
Time Period ◽  
Extreme Flood ◽  
Discrete Uniform Distribution ◽  
L Value ◽  
Discrete Uniform

Abstract. The use of non-systematic flood data for statistical purposes depends on the reliability of the assessment of both flood magnitudes and their return period. The earliest known extreme flood year is usually the beginning of the historical record. Even if one properly assesses the magnitudes of historic floods, the problem of their return periods remains unsolved. The matter at hand is that only the largest flood (XM) is known during whole historical period and its occurrence marks the beginning of the historical period and defines its length (L). It is common practice to use the earliest known flood year as the beginning of the record. It means that the L value selected is an empirical estimate of the lower bound on the effective historical length M. The estimation of the return period of XM based on its occurrence (L), i.e. ^M = L, gives a severe upward bias. The problem arises that to estimate the time period (M) representative of the largest observed flood XM. From the discrete uniform distribution with support 1, 2, ... , M of the probability of the L position of XM, one gets ^L = M/2. Therefore ^M = 2L has been taken as the return period of XM and as the effective historical record length as well this time. As in the systematic period (N) all its elements are smaller than XM, one can get ^M = 2t( L+N). The efficiency of using the largest historical flood (XM) for large quantile estimation (i.e. one with return period T = 100 years) has been assessed using the maximum likelihood (ML) method with various length of systematic record (N) and various estimates of the historical period length ^M comparing accuracy with the case when systematic records alone (N) are used only. The simulation procedure used for the purpose incorporates N systematic record and the largest historic flood (XMi) in the period M, which appeared in the Li year of the historical period. The simulation results for selected two-parameter distributions, values of their parameters, different N and M values are presented in terms of bias and root mean square error RMSEs of the quantile of interest are more widely discussed.

A review on Poisson, Cox, Hawkes, shot-noise Poisson and dynamic contagion process and their compound processes

2020 ◽  
pp. 1-22
Author(s):  
Jiwook Jang ◽  
Rosy Oh
Keyword(s):  
Poisson Process ◽  
Shot Noise ◽  
Value At Risk ◽  
Hawkes Process ◽  
Cox Process ◽  
Doubly Stochastic ◽  
Business Applications ◽  
Tail Conditional Expectation ◽  
Time Period ◽  
Contagion Process

Abstract The Poisson process is an essential building block to move up to complicated counting processes, such as the Cox (“doubly stochastic Poisson”) process, the Hawkes (“self-exciting”) process, exponentially decaying shot-noise Poisson (simply “shot-noise Poisson”) process and the dynamic contagion process. The Cox process provides flexibility by letting the intensity not only depending on time but also allowing it to be a stochastic process. The Hawkes process has self-exciting property and clustering effects. Shot-noise Poisson process is an extension of the Poisson process, where it is capable of displaying the frequency, magnitude and time period needed to determine the effect of points. The dynamic contagion process is a point process, where its intensity generalises the Hawkes process and Cox process with exponentially decaying shot-noise intensity. To facilitate the usage of these processes in practice, we revisit the distributional properties of the Poisson, Cox, Hawkes, shot-noise Poisson and dynamic contagion process and their compound processes. We provide simulation algorithms for these processes, which would be useful to statistical analysis, further business applications and research. As an application of the compound processes, numerical comparisons of value-at-risk and tail conditional expectation are made.

Probabilities of ruin when the safety loading tends to zero

2000 ◽  
Vol 32 (03) ◽  
pp. 885-923 ◽  
Author(s):  
Vsevolod K. Malinovskii
Keyword(s):  
Finite Time ◽  
Absolute Constant ◽  
Classical Result ◽  
Risk Theory ◽  
Premium Rate ◽  
Collective Risk ◽  
Time Period ◽  
Positive Absolute Constant ◽  
Uniform Approximations ◽  
Insurance Business

When the premium rate is a positive absolute constant throughout the time period of observation and the safety loading of the insurance business is positive, a classical result of collective risk theory claims that probabilities of ultimate ruin ψ(u) and of ruin within finite time ψ(t,u) decrease as eϰu with a constant ϰ&gt;0, as the initial risk reserve u increases. This paper establishes uniform approximations to ψ(t,u) with slower rates of decrease when the premium rate depends on u in such a way that the safety loading decreases to zero as u→∞.

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