Derivation of the probability distribution functions for succession quota random variables

Start‐up demonstration tests were first discussed in the quality/reliability literature about three decades ago. Since then, many variations of these tests have been introduced, and the corresponding distributional characteristics and inferential methods have also been studied. All these developments, based on independent and identically distributed binary trials, have been further generalized to some other forms of trials such as Markov‐dependent trials, exchangeable trials and multistate trials. In this paper, we provide a comprehensive review of all these results and highlight some unifications of the results. We also describe a general estimation method and then present several numerical examples to illustrate some of the models and methods described here. Finally, a number of open issues in this area of research are pointed out. Copyright © 2014 John Wiley & Sons, Ltd.

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“…For example, the probability mass function of X , in the i.i.d. binary case, is given by the expression

\begin{matrix} leftalign-star \\ rightalign-odd P MathClass-open(X = n MathClass-close) = A_{s} MathClass-open(n; q, c_{2}, c_{1} MathClass-close) + A_{s} MathClass-open(n; p, c_{1}, c_{2} MathClass-close), for n ⩾ min MathClass-open{c_{1}, c_{2} MathClass-close}, & align-even & rightalign-label \end{matrix}

where

\begin{matrix} leftalign-star \\ rightalign-odd A_{s} MathClass-open(n; q, c_{2}, c_{1} MathClass-close) & align-even = ζ MathClass-open(c_{2}, n MathClass-close) ζ MathClass-open(n, c_{1} + c_{2} - 1 MathClass-close) q^{c_{2}} p^{n - c_{1}} + ζ MathClass-open(c_{2} + 2, n MathClass-close) & rightalign-label & align-label \\ rightalign-odd & align-even \times \sum_{i = 1}^{n - c_{2} - 1} p^{i} q^{n - i} (<...>) \end{matrix}

…”

Section: Binary Start‐up Demonstration Tests Based On Successes and Fmentioning

confidence: 99%

Start‐up demonstration tests: models, methods and applications, with some unifications

Koutras

Milienos

2014

Appl Stoch Models Bus & Ind

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“…Theorem 4.1 and Proposition 3.4 offer an efficient way for the computation of the pmf of W , which is simpler from the respective ways presented by Antzoulakos and Philippou [2,3].…”

Section: Proof It Follows From Theorem 32 Thatmentioning

confidence: 99%

“…Assume now that n ≥ 1. Modifying a combinatorial argument of Philippou and Muwafi [12] (see also Antzoulakos and Philippou [2]), we observe that an arrangement of {0, 1}-sequences of length n belongs to the ones enumerated by A (k,r ) n if and only if it is of the form …”

Section: Sequence Of Extended Fibonacci Numbersmentioning

confidence: 99%

“…Also, let P 1 (n) (P 0 (n)) be the probability that at the nth trial the sooner event between E 1 and E 0 occurs and the sooner event is E 1 (E 0 ). Feller [6, page 327] (see also Ebneshahrashoob and Sobel [5] and Antzoulakos and Philippou [2]) derived the probability generating function of the random variable W which is given by…”

Section: Proof It Follows From Theorem 32 Thatmentioning

confidence: 99%

“…The random variable W has been initially studied by Feller [6, page 327] (see also Ebneshahrashoob and Sobel [5]), and it is closely related to problems of practical interest in statistical theory of runs and reliability theory (see, e.g., Koutras [7], Aki et al [1], and the references therein). The pmf of W is expressed directly in terms of the polynomials A (k,r ) n (x) offering an efficient way for the computation of the pmf of W , which is simpler from the respective ways presented by Antzoulakos and Philippou [2,3]. Relations between the polynomials A (k,r ) n (x) and probability problems which take into account lengths of success and failure runs in a fixed number of n Bernoulli trials are also established.…”

Section: Introduction Let F (K)mentioning

confidence: 99%

See 2 more Smart Citations

Extended Fibonacci numbers and polynomials with probability applications

Antzoulakos

2004

International Journal of Mathematics and Mathematical Sciences

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The extended Fibonacci sequence of numbers and polynomials is introduced and studied. The generating function, recurrence relations, an expansion in terms of multinomial coefficients, and several properties of the extended Fibonacci numbers and polynomials are obtained. Interesting relations between them and probability problems which take into account lengths of success and failure runs are also established.

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Bibliography

2001

Runs and Scans With Applications

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Derivation of the probability distribution functions for succession quota random variables

Abstract: Succession quota, sooner and later problems, Bernoulli trials, binary sequence of order k , probability distribution function, probability generating function, longest run,

Cited by 14 publications

References 16 publications

Start‐up demonstration tests: models, methods and applications, with some unifications

Start‐up demonstration tests: models, methods and applications, with some unifications

Extended Fibonacci numbers and polynomials with probability applications

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