Integrals of geometric random variables involving eight-shaped geodesics on hyperbolic surfaces

The purpose of this article is to present several equivalent characterizations of comparing the largest-order statistics and sample ranges of two sets of n independent exponential random variables with respect to different stochastic orders, where the random variables in one set are heterogeneous and the random variables in the other set are identically distributed. The main results complement and extend several known results in the literature. The geometric distribution can be regarded as the discrete counterpart of the exponential distribution. We also study the orderings of the largest-order statistics from geometric random variables and point out similarities and differences between orderings of the largest-order statistics from geometric variables and from exponential variables.

On the minimum of independent geometrically distributed random variables

Let Sn(a1, …, an) be the sum of n independent exponential random variables with respective hazard rates a1, …, an or the sum of n independent geometric random variables with respective parameters a1, …, an. In this article, we investigate sufficient conditions on parameter vectors (a1, …, an) and $(a_{1}^{\ast},\ldots,a_{n}^{\ast})$ under which Sn(a1, …, an) and $S_{n}(a_{1}^{\ast},\ldots,a_{n}^{\ast})$ are ordered in terms of the increasing convex and the reversed hazard rate orders for both exponential and geometric random variables and in terms of the mean residual life order for geometric variables. For the bivariate case, all of these sufficient conditions are also necessary. These characterizations are used to compare fail-safe systems with heterogeneous exponential components in the sense of the increasing convex and the reversed hazard rate orders. The main results complement several known ones in the literature.

Likelihood ratio ordering of convolutions of heterogeneous exponential and geometric random variables

We use the Stein-Chen method to study the extremal behaviour of univariate and bivariate geometric laws. We obtain a rate for the convergence, to the Gumbel distribution, of the law of the maximum of i.i.d. geometric random variables, and show that convergence is faster when approximating by a discretised Gumbel. We similarly find a rate of convergence for the law of maxima of bivariate Marshall-Olkin geometric random pairs when approximating by a discrete limit law. We introduce marked point processes of exceedances (MPPEs), both with univariate and bivariate Marshall-Olkin geometric variables as marks and determine bounds on the error of the approximation, in an appropriate probability metric, of the law of the MPPE by that of a Poisson process with same mean measure. We then approximate by another Poisson process with an easier-to-use mean measure and estimate the error of this additional approximation. This work contains and extends results contained in the second author’s PhD thesis (Feidt 2013) under the supervision of Andrew D. Barbour.

Let X1,X2 be independent geometric random variables with parameters p1,p2, respectively, and Y1,Y2 be i.i.d. geometric random variables with common parameter p. It is shown that X2:2, the maximum order statistic from X1,X2, is larger than Y2:2, the second order statistic from Y1,Y2, in terms of the hazard rate order [usual stochastic order] if and only if \(p\geq \tilde{p}\), where \(\tilde{p}=(p_{1}p_{2})^{\frac{1}{2}}\) is the geometric mean of (p1,p2). This result answers an open problem proposed recently by Mao and Hu (Probab. Eng. Inf. Sci. 24:245–262, 2010) for the case when n=2.

Gaps in samples of geometric random variables

On the expectation of the maximum of IID geometric random variables

□ We calculate the asymptotics of the moments as well as the limiting distribution (after the appropriate normalization) of the maximum of independent, not identically distributed, geometric random variables. In many cases, the limit distribution turns out to be the standard Gumbel. The motivation comes from a variant of the genomic evolutionary model proposed by Wilf and Ewens[ 15 ] as an answer to the criticism of the Darwinian theory of evolution stating that the time required for the appropriate mutations is huge. A byproduct of our analysis is the asymptotics of the moments as well as the limiting distribution (after the appropriate normalization) of the maximum of independent, not identically distributed, exponential random variables.

Mao and Hu (2010) left an open problem about the hazard rate order between the largest order statistics from two samples of n geometric random variables. Du et al. (2012) solved this open problem when n = 2, and Wang (2015) solved for 2 ≤ n ≤ 9. In this paper we completely solve this problem for any value of n.

Evaluating the influence of setup uncertainties on treatment planning for focal liver tumors

This chapter will discuss various families of discrete random variables and their applications in modeling and validating random events. In this chapter, I will first provide a review of formulas related to discrete random variables in Section 12.1. Then, various kinds of discrete random variables, including Bernoulli, binomial, geometric, Pascal, Poisson, and discrete uniform random variables, will be sequentially presented in Sections 12.2-12.7. Specifically, the probability mass function (pmf), expected value, variance, and properties of each of these random variables will be derived and discussed in detail.

Chapter 3 - Probability Distributions

Evaluating the influence of setup uncertainties on treatment planning for focal liver tumors

Schur properties of convolutions of exponential and geometric random variables