Successes,runs and longest runs |
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| Affiliation: | 1. School of Civil and Environmental Engineering, Georgia Institute of Technology, Atlanta, United States;1. Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET)-Delegación Técnica Patagonia, Administración de Parques Nacionales, O''Connor 1180, 8400 San Carlos de Bariloche, Río Negro, Argentina;2. Proyecto Macá Tobiano – Aves Argentinas/Asociación Ornitológica del Plata y Asociación Ambiente Sur, Argentina;3. Departamento de Ecología, Genética y Evolución - Instituto de Ecología, Genética y Evolución de Buenos Aires (IEGEBA), Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET), Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Pabellón II Ciudad Universitaria, Ciudad Autónoma de Buenos Aires C1428EGA, Argentina;4. Zoological Society of London, United Kingdom;1. DST/NRF Research Chair in Shallow Water Ecosystems, Nelson Mandela Metropolitan University, Port Elizabeth 6031, South Africa;2. Department of Botany and the Coastal and Marine Research Institute, Nelson Mandela Metropolitan University, Port Elizabeth 6031, South Africa;3. South African Environmental Observation Network, Elwandle Coastal Node, Port Elizabeth 6031, South Africa |
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| Abstract: | The probability distribution of the numbeer of success runs of length k ( >/ 1) in n ( ⩾ 1) Bernoulli trials is obtained. It is noted that this distribution is a binomial distribution of order k, and several open problems pertaining to it are stated. Let Sn and Ln, respectively, denote the number of successes and the length of the longest success run in the n Bernoulli trials. A formula is derived for the probability P(Ln ⩽ k | Sn = r) (0 ⩽ k ⩽ r ⩽ n), which is alternative to those given by Burr and Cane (1961) and Gibbons (1971). Finally, the probability distribution of Xn, Ln(k) is established, where Xn, Ln(k) denotes the number of times in the n Bernoulli trials that the length of the longest success run is equal to k. |
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