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A noisy duel is a zero-sum, two-person game with the following structure. Each player has bullets which he can fire at any time in [0, 1]. If Player i fires at time t, he hits with probability Pi(t). The functions Pi are continuous and nondecreasing with P[sub i](0) = 0 and P[sub i](l) = 1. The number of bullets each player possesses at any time and the functions Pi are known to both. The payoff is I to the sole survivor, otherwise 0. This article states the authors' earlier result on the existence of a value of a noisy duel and presents a detailed discussion of the structure of epsilon-good strategies and criteria for the existence of good first-shot times. We present tables of values and shooting times for noisy duels, which, in some cases, can be used to trace the play of the game. An additional table illustrates how large an arsenal is necessary to overcome the effects of an opponent's superior accuracy. [ABSTRACT FROM AUTHOR]

Cochran's Q test for differences between related-sample percentages or proportions has generally been incorrectly presented in secondary sources. The most common mistake results from failure to recognize that rows containing only 1's or only O's, i.e., only successes or only failures, do not affect the value of Q. The F test, however, is affected by such rows. The probabilities from the x[sup 2] and F approximations are compared with the exact probabilities in three sets of data. A rule of thumb, based on extensive study of the distribution of Q in small samples, is given as an aid in judging when the x[sup 2] approximation is satisfactory for practical purposes. [ABSTRACT FROM AUTHOR]

Results are developed for play-the-winner sampling for choosing the better of two binomial populations when the maximum number of tests is specified. It is shown that for a fixed number of tests the probability of correct selection with alternating assignment exceeds that for play- the-winner sampling. However, when the probability of correct selection is fixed, neither sampling method is uniformly better than the other as measured by the expected number of tests on the poorer population, or on the total expected number of tests. [ABSTRACT FROM AUTHOR]

We investigate the problem of finding the expected value of functions of a random variable X of the form f(X) = (X + A)[sup -n] where X + A > 0 a.s. and n is a non-negative integer. The technique is to successively integrate the probability generating function and is suggested by the well-known result that successive differentiation leads to the positive moments. The technique is applied to the problem of finding E[1/(X + A)] for the binomial and Poisson distributions. [ABSTRACT FROM AUTHOR]