Common Probability Distribution Research Articles

On relative effectiveness in duels

We study the relative effectiveness of two multi-unit forces, A and B, fighting a successive series of duels between randomly selected pairs of opposing units. A duel consists of a series of games - each of at most s units of time - and the duel terminates as soon as one of the two duelists is hit. A new independent duel then starts, and so on. Each force is characterized by a common probability distribution function representing the time required by a single unit of the force to hit a non-firing unit of the other force in a game of unbounded duration. The relative effectiveness of force A with respect to force B is expressed by the exchange rateR(s), defined as the expected number of units of force B hit by a single unit of force A until unit A itself is hit. We derive a general characterization of R{s) and develop analytic results for several choices of pairs of families of time-to-hit distributions. We show that the shape of the distributions can strongly affect the outcome of the duels and that this e...

The Effect of Sample Design on Principal Component Analysis

Abstract Most sample surveys are multivariate and many lend themselves to multivariate methods of analysis. The most usual mode of such analysis is a standard statistical package, such as BMDP or SPSS, in which the multivariate analyses are based on the underlying assumption that the data are generated as independent observations from a common probability distribution. This assumption ignores the sample selection procedure involved in the survey, which leads to the following basic questions. What effects can the sample design have on methods of multivariate analysis? How should such effects be taken into account? This article considers the case of principal component analysis and, in particular, the point estimation of the eigenvalues and eigenvectors of a covariance matrix. It is assumed that the selection of the sample depends on the population values of auxiliary variables as, for example, in stratified sampling. The conventional estimators, based on the assumption of simple random sampling, are compar...

A theorem for physicists in the theory of random variables

A theorem is presented which tells how to calculate the joint probability distribution of m random variables that have been defined as functional transformations of n given random variables (m, n≥1). Although the theorem involves Dirac delta functions and therefore has a rather formal appearance, it turns out to be surprisingly useful. It is used here to develop a number of important results in random variable theory, including: the central limit theorem, the density functions of some common probability distributions (lognormal, chi-square, and Student’s-t), the imposition of constraints on random variables, some sampling properties of the normal distribution, and the chi-square goodness-of-fit test. Special attention is given to aspects of these topics that find fruitful application in physics. A broad conclusion seems to be that the theorem presented here provides a systematic way of obtaining many results in random variable theory which, although quite useful in physics, are normally found derived only in moderately advanced mathematics textbooks.

On the observation closest to the origin

Out of n i.i.d. random vectors in R d let X ∗ n be the one closest to the origin. We show that X ∗ n has a nondegenerate limit distribution if and only if the common probability distribution satisfies a condition of multidimensional regular variation. The result is then applied to a problem of density estimation.

A Necessary and Sufficient Condition for Reaching a Consensus Using DeGroot's Method

Abstract DeGroot (1974) proposed a model in which a group of k individuals might reach a consensus on a common subjective probability distribution for an unknown parameter. This paper presents a necessary and sufficient condition under which a consensus will be reached by using DeGroot's method. This work corrects an incorrect statement in the original paper about the conditions needed for a consensus to be reached. The condition for a consensus to be reached is straightforward to check and yields the value of the consensus, if one is reached.

Coincidence probabilities for simple majority and positional voting rules

With three candidates and an odd number n of voters, let Q( n, λ, p) be the probability that the winning candidate under the point-total rule that assigns 1, λ, and 0 points respectively to each first, second, and third-place vote is the same as the simple majority candidate, given that there exists a simple majority candidate, when each voter independently selects a linear preference order on the candidates by a common probability distribution p on the six linear orders on the candidates. With Q( n, λ) = Q( n, λ, p) when p assigns probability 1 6 to each order, the λ values that maximize Q( n, λ) for small n consist of open intervals in [0, 1 2 ]. Using quadrivariate normals, a computational form is developed for the limiting probability Q( λ) = lim n→∞ Q( n, λ). The function Q( λ) = Q(1 − λ) for each λ ϵ [0, 1] and is differentiable with Q( λ) strictly increasing as λ goes from 0 to 1 2 . The maximum value Q( 1 2 ) is approximately. 901189. Effects of nonuniform p distributions on Q( n, λ, p) are also discussed.

Confidence limits for design events

Frequency analysis is commonly used in hydrology to define flood plains, design hydraulic structures, and aid in watershed planning and management. Although design event magnitudes may be determined analytically by using probability distributions, the analytical determination of confidence limits for the design event is not easy. Using the alternate method of moments to determine confidence limits involves an assumption of normality of the distribution of design events. It is shown by data generation that this assumption is valid for quite small samples. Tables are given to compute confidence limits in this manner for several common probability distributions.

Reaching a Consensus

Abstract Consider a group of individuals who must act together as a team or committee, and suppose that each individual in the group has his own subjective probability distribution for the unknown value of some parameter. A model is presented which describes how the group might reach agreement on a common subjective probability distribution for the parameter by pooling their individual opinions. The process leading to the consensus is explicitly described and the common distribution that is reached is explicitly determined. The model can also be applied to problems of reaching a consensus when the opinion of each member of the group is represented simply as a point estimate of the parameter rather than as a probability distribution.

Two Instructive Examples of Maximum Likelihood Estimates

In some of the more common probability distributions such as the Bernoulli, the Poisson, and ihe Gaussian where the parameter (or one of the parameters) is the mean of the distribution (or of the random variable having that distribution), it happens that the maximum likelihood estimate (m. 1. e.) of that parameter is the sample mean. In still other distributions such as the binomial, a simple change in one of the parameters will produce this result. For example, if in the binomial distribution with para. meters n and p. we replace p by 0/n, then 0 is the mean of the distribution and the m. 1. e. of 0 is the sample mean. Following are two examples, one for the discrete case and one for the continuous case, which show that it is possible to have distributions in which the parameter is the mean of the distribution bu-t for which the m. 1. e. of that parameter is not the sample mean. For the discrete case, we might let X be a random variable with the probability function

Exact Distribution of the Sum of Independent Identically Distributed Discrete Random Variables

Abstract Let Sn be the sum of n independent discrete variates with common probability distribution px (x = 0, 1, 2, [mdot][mdot][mdot]). A table is presented which allows Pr (Sn = s) to be written down, for any n and for s ≤15, as a finite series involving mixed powers of the px 's. If the range of x is finite (x ≤ m) the table will also give Pr(Sn = mn - s). The use of the table is illustrated and several applications are indicated.

Exact Distribution of the Sum of Independent Identically Distributed Discrete Random Variables

Abstract Let Sn be the sum of n independent discrete variates with common probability distribution px (x = 0, 1, 2, [mdot][mdot][mdot]). A table is presented which allows Pr (Sn = s) to be written down, for any n and for s ≤15, as a finite series involving mixed powers of the px 's. If the range of x is finite (x ≤ m) the table will also give Pr(Sn = mn - s). The use of the table is illustrated and several applications are indicated.