{"title":"伯努利试验成功运行概率的近似","authors":"S. Kaczkowski","doi":"10.1090/tpms/1186","DOIUrl":null,"url":null,"abstract":"Concise and convenient bounds are obtained for the probability mass and cumulative distribution functions associated with the first success run of length \n\n \n k\n k\n \n\n in a sequence of \n\n \n n\n n\n \n\n Bernoulli trials. Results are compared to an approximation obtained by the Stein–Chen method as well as to bounds obtained from statistical reliability theory. These approximation formulas are used to obtain precise estimates of the expectation value associated with the occurrence of at least one success run of length \n\n \n k\n k\n \n\n within \n\n \n N\n N\n \n\n concurrent sequences of Bernoulli trials.","PeriodicalId":42776,"journal":{"name":"Theory of Probability and Mathematical Statistics","volume":null,"pages":null},"PeriodicalIF":0.4000,"publicationDate":"2023-05-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Approximations for success run probabilities in Bernoulli trials\",\"authors\":\"S. Kaczkowski\",\"doi\":\"10.1090/tpms/1186\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Concise and convenient bounds are obtained for the probability mass and cumulative distribution functions associated with the first success run of length \\n\\n \\n k\\n k\\n \\n\\n in a sequence of \\n\\n \\n n\\n n\\n \\n\\n Bernoulli trials. Results are compared to an approximation obtained by the Stein–Chen method as well as to bounds obtained from statistical reliability theory. These approximation formulas are used to obtain precise estimates of the expectation value associated with the occurrence of at least one success run of length \\n\\n \\n k\\n k\\n \\n\\n within \\n\\n \\n N\\n N\\n \\n\\n concurrent sequences of Bernoulli trials.\",\"PeriodicalId\":42776,\"journal\":{\"name\":\"Theory of Probability and Mathematical Statistics\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.4000,\"publicationDate\":\"2023-05-02\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Theory of Probability and Mathematical Statistics\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1090/tpms/1186\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q4\",\"JCRName\":\"STATISTICS & PROBABILITY\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Theory of Probability and Mathematical Statistics","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1090/tpms/1186","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"STATISTICS & PROBABILITY","Score":null,"Total":0}
Approximations for success run probabilities in Bernoulli trials
Concise and convenient bounds are obtained for the probability mass and cumulative distribution functions associated with the first success run of length
k
k
in a sequence of
n
n
Bernoulli trials. Results are compared to an approximation obtained by the Stein–Chen method as well as to bounds obtained from statistical reliability theory. These approximation formulas are used to obtain precise estimates of the expectation value associated with the occurrence of at least one success run of length
k
k
within
N
N
concurrent sequences of Bernoulli trials.