Download Geometric Sums: Bounds for Rare Events with Applications: by Vladimir V. Kalashnikov PDF

By Vladimir V. Kalashnikov

This e-book reports difficulties linked to infrequent occasions coming up in quite a lot of situations, treating such subject matters as tips to evaluation the chance an assurance corporation can be bankrupted, the life of a redundant procedure, and the ready time in a queue.
Well-grounded, certain mathematical evaluate equipment of uncomplicated chance features fascinated with infrequent occasions are offered, which might be hired in actual purposes, because the quantity additionally includes appropriate numerical and Monte Carlo tools. a number of the examples, tables, figures and algorithms can also be preferred.
Audience: This paintings may be worthy to graduate scholars, researchers and experts drawn to utilized likelihood, simulation and operations research.

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Extra resources for Geometric Sums: Bounds for Rare Events with Applications: Risk Analysis, Reliability, Queueing

Example text

23) independent of X n , n ~ 1. Less evidently that the probability q in this case is equal to pl(l + p). In addition, it can be proved that F(u) = P(X ~ u) = AJ u v dB(v). 49). 3 are unified by certain common features of the mathematical model used to describe them and the techniques used to investigate them. With this in mind, let us consider the following setup. Let X I ,X 2 , ... f. F(x). v. which is independent of this sequence and has the geometric distribution P(ZJ=k)=q(l-q)k-l, k>l. 1) Denote Sv Wq(x) = Xl + ...

Y. By this, the inequality = I Ef(X) - = Ef(Y) I :::; (s(X, Y) Miscellaneous Probability Topics 35 o yields the corollary. We now state facts that are concerned with the types of convergence induced by probability metrics. 's having distributions F n, n 2: 0, and discuss different types of convergence of X n (when n-+=) to X 0 or, more precisely, convergence of F n to F o. The total variation metric induces the strongest type of convergence which is called the strong convergence or the total variation convergence.

Evidently, E I X I s < 00 ¢} Ef(X) < 00 and the same is true for r. v. Y. By this, the inequality = I Ef(X) - = Ef(Y) I :::; (s(X, Y) Miscellaneous Probability Topics 35 o yields the corollary. We now state facts that are concerned with the types of convergence induced by probability metrics. 's having distributions F n, n 2: 0, and discuss different types of convergence of X n (when n-+=) to X 0 or, more precisely, convergence of F n to F o. The total variation metric induces the strongest type of convergence which is called the strong convergence or the total variation convergence.

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