The Gaussian polytope $\mathcal P_{n,d}$ is the convex hull of n independent standard normally distributed points in $\mathbb{R}^d$ . We derive explicit expressions for the probability that $\mathcal P_{n,d}$ contains a fixed point $x\in\mathbb{R}^d$ as a function of the Euclidean norm of x, and the probability that $\mathcal P_{n,d}$ contains the point $\sigma X$ , where $\sigma\geq 0$ is constant and X is a standard normal vector independent of $\mathcal P_{n,d}$ . As a by-product, we also compute the expected number of k-faces and the expected volume of $\mathcal P_{n,d}$ , thus recovering the results of Affentranger and Schneider (Discr. and Comput. Geometry, 1992) and Efron (Biometrika, 1965), respectively. All formulas are in terms of the volumes of regular spherical simplices, which, in turn, can be expressed through the standard normal distribution function $\Phi(z)$ and its complex version $\Phi(iz)$ . The main tool used in the proofs is the conic version of the Crofton formula.

We consider birth-death processes taking values in but allow the death rate in state 0 to be positive, so that escape from is possible. Two such processes with transition functions are said to be similar if, for all there are constants cij such that for all t ≥ 0. We determine conditions on the birth and death rates of a birth-death process for the process to be a member of a family of similar processes, and we identify the members of such a family. These issues are also resolved in the more general setting in which the two processes are called similar if there are constants cij and ν such that for all t ≥ 0.

This paper considers the absorption of a non-decreasing compound Poisson process of finite order in a general upper boundary. The problem is relevant in fields such as risk theory, Kolmogorov–Smirnov statistics and sequential analysis. The probability of absorption and first-passage times are given in terms of a generating function which depends on the boundary only and can be computed readily. Absorption is certain or not as the asymptotic slope of the boundary is greater or less than the expected increase of the process in unit time. The case of the linear boundary is considered in detail.

We prove a conjecture of Arnold (1968) which simplifies the determination of an optimal bound on absorption probability originally due to Moran (1960).

Vehicles whose lengths are independent identically distributed random variables with known distribution function are loaded onto ferries of fixed known length, each ferry departing as soon as it can no longer accommodate the next vehicle in the queue. We work out how much a vehicle of any particular length ought to pay for use of the ferry, as well as the expected number of vehicles per ferry and expected revenue per ferry in equilibrium.

Let {W(t), 0≦t<∞} be the standard Wiener process. The computation schemes developed in the past are not computationally efficient for the absorption probabilities of the type P{sup0≦t≦TW(t) − f(t) ≧ 0} when either T is large or f(0) > 0 is small. This paper gives an efficient and accurate algorithm to compute such probabilities, and some applications to other Gaussian stochastic processes are discussed.