Independent Discrete Random Variables Research Articles

Probabilistic location problems with discrete demand weights

AbstractIn this article we consider four models for locating a facility on an undirected network with demand weights, which are independent discrete random variables. These problems include the probabilistic versions of two models for locating desirable facilities: the 1‐median and 1‐minimax problems and two problems for locating undesirable facilities: the 1‐antimedian and 1‐maximin problems. The article contains analysis of special cases where a solution is determined by solving deterministic versions of the problems and efficient algorithms to solve the problems in general. © 2004 Wiley Periodicals, Inc. NETWORKS, Vol. 44(1), 47–57 2004

A HIGHER ORDER LARGE-DEVIATION APPROXIMATION FOR THE DISCRETE DISTRIBUTIONS

For a sum of independent discrete random variables, its higher order largedeviation approximation is discussed. An approximation to the tail probability of the distribution of the sum is provided, and its numerical comparison with other approximations is done in the binomial case. Consequently, the approximation formula is seen to be more accurate.

Acceptance–Rejection Sampling from the Conditional Distribution of Independent Discrete Random Variables, given their Sum

The problem of drawing samples from the conditional distribution of a vector of independent, but not necessarily identically distributed, discrete random variables, given their sum, is considered. The traditional method of acceptance–rejection sampling can be impossible in practice if the statistic to condition upon has an extremely low probability of assuming the conditioning value. It is shown how the sufficiency principle combined with tilting of distributions may be used to find an equivalent problem, where the conditioning event has a much higher probability to occur. As applications, a test for proportional hazards in survival analysis and a test of no trend in a logit model with offset are given.

The higher order large-deviation approximation for the distribution of the sum of independent discreterandom variables

For a sum of not identic ally but independently distributed discrete random variables, its higher order large-deviation approximation in given. They are compared with the normal and Edge-worth type approximations in various cases. Consequently, the large-deviation approximations give sufficiently accurate results.

Using Spreadsheets to Calculate Prob (X+Y=W)

SummaryUsing the spreadsheet function SUMPRODUCT which calculates dot products, it is possible to compute the probabilities associated with the sums of independent discrete random variables. This allows the student to confirm properties of the sum of binomial and Poisson random variables. It also provides a method for computing the distribution of the sum of two (or more) arbitrary random variables. In addition, it also allows the student to compute the probabilities associated with the difference of random variables, and thus find the probability that one random variable exceeds another (or exceeds by a given amount).

On the exact convolution of discrete random variables

We present two expressions for the exact density function for the sum of independent discrete random variables. Application to binomial random variables is illustrated. Evaluation of the exact density is practical on the personal computer. Extension to other discrete random variables is straightforward.

Further paradoxes in admissibility

Let X, Y be independent discrete random variables distributed according to exponential families d P θ ( x)=e θx β( θ) μ( x) θ ϵ Λ, d Q Ψ ( y)=e yΨ γ( Ψ) ν( y) Ψ ϵ Γ, respectively. We assume Λ and Γ are the natural parameter spaces of X and Y. This paper is concerned with the estimation of any given function g( θ) in a fairly general class of functions. We will consider estimators of the form δ( Y) and given conditions under which these estimators are admissible for estimating g( θ) under a certain standard class of loss function; let L denote one of these loss functions. The anomaly is that these estimators δ( Y) are admissible for estimating g( θ) but do not depend on the parameter θ. In particular it is shown that if the sample space T of Y is bounded below or above then all estimators δ( Y) taking values in R ∗ = [g(θ):θ ϵ Λ] and also in certain boundary points for R ∗ are admissible for estimating g( θ) under loss L. It should be pointed out that the boundary points mentioned above depend on the distribution of X. More generally we show that if the natural parameter space Γ (for Y) is unbounded below (respectively above) then all estimators δ( Y) are admissible for estimating g( θ) under loss L provided that: 1. (a) the range of values of δ( Y) is in R ∗ ; and 2. (b) δ( Y) takes on some constant value c ϵ R ∗ as y → −∞ (respectively + ∞). Extensions of the above results to non-discrete and/or non-exponential families are briefly discussed.

Wald's identity and absorption probabilities for two-dimensional random walks

SummarySuppose a particle executes a random walk on a two-dimensional square lattice, starting at the origin. The position of the particle after n steps of the walk is Xn = (Xl, n, X2n), whereand we will assume that the Yi are independent bivariate discrete random variables with common moment generating function (m.g.f.)where a, b, c and d are non-negative. We assume further that (i) pi, j is non-zero for some finite positive and negative i, and some finite positive and negative j (− a ≤ i ≤ b, − c ≤ jd), such values of i and j including – a, b and – c, d, respectively, whenever a, b, c or d is finite, and (ii) the double series defining Φ(α, β) is convergent at least in some finite region D, of the real (α, β) plane, which includes the origin.