Boundaries from Inhomogeneous Bernoulli Trials

AbstractThis paper is devoted to random-bit simulation of probability densities, supported on {[0,1]}. The term “random-bit” means that the source of randomness for simulation is a sequence of symmetrical Bernoulli trials. In contrast to the pioneer paper [D. E. Knuth and A. C. Yao, The complexity of nonuniform random number generation, Algorithms and Complexity, Academic Press, New York 1976, 357–428], the proposed method demands the knowledge of the probability density under simulation, and not the values of the corresponding distribution function. The method is based on the so-called binary decomposition of the density and comes down to simulation of a special discrete distribution to get several principal bits of output, while further bits of output are produced by “flipping a coin”. The complexity of the method is studied and several examples are presented.

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n-point, win-by-k games

Journal of Applied Probability ◽

10.2307/3214385 ◽

1989 ◽

Vol 26 (4) ◽

pp. 807-814 ◽

Cited By ~ 7

Author(s):

Kyle Siegrist

Keyword(s):

Exit Time ◽

Bernoulli Trials ◽

Expected Number ◽

First Exit Time ◽

World Series ◽

Time Space ◽

Gambler’S Ruin ◽

Gambler's Ruin Problem ◽

Positive Integers ◽

Made In

Consider a sequence of Bernoulli trials between players A and B in which player A wins each trial with probability p∈ [0, 1]. For positive integers n and k with k ≦ n, an (n, k) contest is one in which the first player to win at least n trials and to be ahead of his opponent by at least k trials wins the contest. The (n, 1) contest is the Banach match problem and the (n, n) contest is the gambler's ruin problem. Many real contests (such as the World Series in baseball and the tennis game) have an (n, 1) or an (n, 2) format. The (n, k) contest is formulated in terms of the first-exit time of the graph of a random walk from a certain region of the state-time space. Explicit results are obtained for the probability that player A wins an (n, k) contest and the expected number of trials in an (n, k) contest. Comparisons of (n, k) contests are made in terms of the probability that the stronger player wins and the expected number of trials.

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Some Elementary Results on Poisson Approximation in a Sequence of Bernoulli Trials

SIAM Review ◽

10.1137/1020070 ◽

1978 ◽

Vol 20 (3) ◽

pp. 567-579 ◽

Cited By ~ 55

Author(s):

R. J. Serfling

Keyword(s):

Poisson Approximation ◽

Bernoulli Trials

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A Systematic Framework for Composite Hypothesis Testing of Independent Bernoulli Trials

IEEE Signal Processing Letters ◽

10.1109/lsp.2015.2395811 ◽

2015 ◽

Vol 22 (9) ◽

pp. 1249-1253 ◽

Cited By ~ 53

Author(s):

D. Ciuonzo ◽

A. De Maio ◽

P. Salvo Rossi

Keyword(s):

Hypothesis Testing ◽

Bernoulli Trials ◽

Composite Hypothesis ◽

Systematic Framework

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Replicating sequences of bernoulli trials in simulation modeling

Computers & Operations Research ◽

10.1016/0305-0548(76)90018-6 ◽

1976 ◽

Vol 3 (4) ◽

pp. 357-361

Author(s):

Bertram Price

Keyword(s):

Simulation Modeling ◽

Bernoulli Trials

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A Note on a Lower Bound for the Multiplicative Odds Theorem of Optimal Stopping

Journal of Applied Probability ◽

10.1017/s0021900200011748 ◽

2014 ◽

Vol 51 (03) ◽

pp. 885-889 ◽

Cited By ~ 1

Author(s):

Tomomi Matsui ◽

Katsunori Ano

Keyword(s):

Lower Bound ◽

Optimal Stopping ◽

Secretary Problem ◽

Bernoulli Trials ◽

Optimal Stopping Problem ◽

Maximum Probability ◽

Optimal Stopping Rule ◽

Optimal Probability ◽

Optimal Stopping Theory ◽

Special Case

In this note we present a bound of the optimal maximum probability for the multiplicative odds theorem of optimal stopping theory. We deal with an optimal stopping problem that maximizes the probability of stopping on any of the last m successes of a sequence of independent Bernoulli trials of length N, where m and N are predetermined integers satisfying 1 ≤ m < N. This problem is an extension of Bruss' (2000) odds problem. In a previous work, Tamaki (2010) derived an optimal stopping rule. We present a lower bound of the optimal probability. Interestingly, our lower bound is attained using a variation of the well-known secretary problem, which is a special case of the odds problem.

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