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The lack of memory property of the exponential distribution plays an important part in the branch of applied probability. This property assumes the information regarding the probability distribution. In the present paper we give a characteristic property of the exponential distribution based on the means of the conditional distributions. Considering a random variable T with finite mean and such that P(T > O) > 0, and denoting by y a positive number such that P(T > y) > 0, we show that T has an exponential distribution if and only if the mean of the conditional distribution, given T > y, exceeds the mean of the unconditional distribution by the quantity y for all y. Also given in the paper is a similar characterization for the geometric distribution. Since the information concerning the expected values is easily accessible, we expect these properties to be useful in dealing with the practical problems. [ABSTRACT FROM AUTHOR]

The principal part of this paper is devoted to the study of the distribution and density functions of the ratio of two normal random variables. It gives several representations of the distribution function in terms of the vivariate normal distribution and Nicholson's V function, both of which have been extensively studied, and for which tables and computational procedures are readily available. One of these representations leads to an easy derivation of the density function in terms of the Cauchy density and the normal density and integral. A number of graphs of the possible shapes of the density are given, together with an indication of when the density is unimodal or bimodal. The last part of the paper discusses the distribution of the ratio (u[sub 1] + ... + u[sub n])/(v[sub I] + ... + v[sub m]) where the u's and v's are independent, uniform variables. The exact distribution for all n and m is given, and some approximations discussed. [ABSTRACT FROM AUTHOR]

In two papers by C. J. ANCKER, JR., A. F. GAFARIAN, AND R. K. GRAY, the structure of the headways at the passage of an oversaturated signalized intersection were studied. The statistical analysis performed by these authors seems to confirm the hypothesis that these headways are independent random variables. In this paper we consider the following simple model. Let the spacings and reaction times of the vehicles stopped at the intersection all be independent and let the trajectory relative to the start position and start time be the same for all stopped vehicles. It is shown that the head- ways at the intersection are independent if and only if this trajectory is linear in a certain region specified in the paper. [ABSTRACT FROM AUTHOR]

In 1965 Stanley Warner [8] illustrated a technique whereby one could estimate from a sample the proportion of persons in a population possessing some characteristic X, without pointedly asking the question, "Do you possess characteristic X?". The present article uses a variation of these ideas suggested by Warner in a later paper [7] to estimate the distribution function of a continuous-type random variable. The technique is illustrated by estimating an income distribution from the responses of a sample of 500 individuals. The potential use of devices of this type in maintaining confidentiality of existing data files is apparent. [ABSTRACT FROM AUTHOR]

This paper uses a family of bivariate extreme-value distributions to estimate the probability of a large exceedance of a random variable given that a certain other random variable not independent of the first has exceeded a certain value. A simple method of reasonably good efficiency is given for estimating a bivariate extreme-value distribution from independent bivariate samples. The method is used to analyze the performance of a spacecraft command receiver which has an indication of data quality so that commands likely to be in error can be rejected. [ABSTRACT FROM AUTHOR]

The purpose of the paper is to illustrate a "give and take" type of interaction between the frequentist theory of probability and statistics, on the one hand, and research in science, on the other. Invariably, research in science involves some observations x and the unknown "true" state of nature sigma. Quite often, replication of the experiments reveals considerable variation in x, indicating that no mathematical treatment of the problem is possible without the assumption that x is a sample value of a random variable X, treated in terms of frequentist theory of probability. As to the true state of nature, sigma, situations vary. Indeed, there are cases where it appears natural to consider that sigma is selected at random out of a certain known set SIGMA, with either known or unknown probability law. Section 2 lists three typical examples in which the frequentist theory of probability can "give" something to science. In each example, however, the assumption of the randomness of sigma appears extraneous. Section 3 describes another example of research in science, again with a non-random sigma, which happened to "give" something to the frequentist theory of statistics. The motivation to reduce the frequency of erroneous conclusions in the general circumstances of the biological study had led to considering the possibility that a certain derivative may fail to exist. This created a problem of testing (Section 4) which does not appear to have been considered. A solution of the problem, in the form of optimal symmetric (C alpha) tests, is given in Section 5. [ABSTRACT FROM AUTHOR]

Let X[sub (1)] is less than or equal to X[sub (2] is less than or equal to ... is less than or equal to X[sub (n)] be an ordered sample of a random variable X which has continuous probability distribution function F (x), and let F[sub n] (x) be the corresponding empirical distribution function. The following three statistics, introduced by A. Renyi, are considered: [Multiple line equation(s) cannot be represented in ASCII text] The paper presents table of exact probabilities for these statistics for finite sample sizes. The limiting distributions of these statistics for sample size n arrow right Infinity are discussed, and sample sizes are indicated for which these limiting distributions can be used instead of the exact distributions. Numerical examples for the use of the tables are presented, as well as applications to testing hypotheses on life distributions and to one-sided estimation of probability distribution functions from censored data. [ABSTRACT FROM AUTHOR]

This paper deals with certain aspects of variance-component estimation for the unbalanced two-way completely-random classification where the numbers of observations in the subclasses are treated as random variables not necessarily independent of some of the random effects of the model. General results are given on the expectations of two commonly-used estimators of the vector of variance components. Numerical approximations are presented for these expectations for one sub-family of the family of all possible joint distributions of the subclass numbers and the random effects. [ABSTRACT FROM AUTHOR]

This paper deals with certain aspects of variance component estimation for the unbalanced one-way random classification where the number (N[sub I]) of observations in the ith class is treated as a random variable not necessarily independent of the class effect (A[sub iota]). It is assumed that in general P(N[sub I] = 0) > 0. The conditional expectations (given the number of observations in each class) of all estimators of the between variance component (sigma[sup 2, sub alpha]) belonging to a certain class of estimators are derived. A general expression is found for the expected value of that estimator of sigma[sup 2, sub alpha] yielded by analysis of variance of class means. The limit of this expression (as the number of classes arrow right Infinity) is given; and it is shown that, if the bivariate distribution function of A[sub I], N[sub I] belongs to a certain class of distribution functions, then this limit is less than sigma[sup 2, sub a]. Numerical approximations to the expected values of two estimators of sigma[sup 2, sub a] are presented for one subclass of such distribution functions. [ABSTRACT FROM AUTHOR]

Tins paper illustrates, through a special case, an effect of non-independent numerators on F-test's in an analysis of variance. Symmetrical balanced incomplete block designs are considered with "blocks" a random effect and no interaction. The joint density of the adjusted sums of squares is determined under the null hypothesis of no treatment effect. Given the result of testing for the block effect, the conditional probability of a Type I error when testing the null hypothesis concerning the treatment effect is evaluated for a number of designs. [ABSTRACT FROM AUTHOR]

This paper considers some problems in the probabilistic analysis of physical evidence in criminal investigations. Two basic assumptions are made: (1) That the number of persons or objects possessing a particular set of properties can be considered as a random variable, and (2) that it is possible to estimate the probability function of this random variable. Two models (one with and one without the assumption that the suspect is a random selection from the set of possible suspects) which arc applicable to the evaluation of partial transfer evidence--an important category of physical evidence that is found in most criminal investigations--are developed. A detailed analysis of the models and an example are presented for the case when the estimated probability distribution is binomial with an expected value less than 1. As the expected value becomes smaller, the assumption of randomness in the selection of the suspect becomes immaterial to the evaluation of the evidential significance. [ABSTRACT FROM AUTHOR]