In renewal theory, the Inspection Paradox refers to the fact that an interarrival period in a renewal process which contains a fixed inspection time tends to be longer than one for the corresponding uninspected proces...In renewal theory, the Inspection Paradox refers to the fact that an interarrival period in a renewal process which contains a fixed inspection time tends to be longer than one for the corresponding uninspected process. We focus on the paradox for Bernoulli trials. Probability distributions and moments for the lengths of the interarrival periods are derived for the inspected process, and we compare them to those for the uninspected case.展开更多
In several instances of statistical practice, it is not uncommon to use the same data for both model selection and inference, without taking account of the variability induced by model selection step. This is usually ...In several instances of statistical practice, it is not uncommon to use the same data for both model selection and inference, without taking account of the variability induced by model selection step. This is usually referred to as post-model selection inference. The shortcomings of such practice are widely recognized, finding a general solution is extremely challenging. We propose a model averaging alternative consisting on taking into account model selection probability and the like-lihood in assigning the weights. The approach is applied to Bernoulli trials and outperforms Akaike weights model averaging and post-model selection estimators.展开更多
Consider performing a sequence of Bernoulli trials (each resulting in either a success, denoted S, or a failure F, with a probability of p and q := 1 - p respectively) until one of m specific strings (or patterns) of ...Consider performing a sequence of Bernoulli trials (each resulting in either a success, denoted S, or a failure F, with a probability of p and q := 1 - p respectively) until one of m specific strings (or patterns) of consecutive outcomes is generated. This can be seen as a game where m players select one such pattern each and the one whose pattern occurs first wins. We present symbolic formulas for the m probabilities of winning, and for the mean number of trials and the corresponding standard deviation to complete this game. Several numerical examples are presented, including a search for optimal strategy.展开更多
A probability forecast method of earthquake magnitude, based on the earthquake frequency magnitude relation and the model of Bernoulli′s random independent trial, is applied to the earthquake risk assessmen...A probability forecast method of earthquake magnitude, based on the earthquake frequency magnitude relation and the model of Bernoulli′s random independent trial, is applied to the earthquake risk assessment of seismic zones in China's Mainland before A.D.2005 in the paper. The forecasting results indicate that the probabilities of earthquake occurrence with magnitude 5 in seismic zones before 2005 are estimated to be over 0.7 in common and 0.8 in most zones; and from 0.5 to 0.7 with M =6; the maximum probability of earthquake occurrence with magnitude 7 is estimated at 0.858, which is also expected in Shanxi seismic zone. In west China's Mainland, earthquakes with magnitude 6 are expected to occur in most seismic zones with high probability (over 0.9 in general) ; the relatively high probabilities of earthquake occurrence (more than 0.7) with magnitude 7 are expected in the seismic zones surrounding the Qinghai Tibet plateau and south Tianshan seismic zone. A discussion about the result confidence and the relationship between the estimated probability and the possible annual rate of earthquake occurrence is made in the last part of the paper.展开更多
Consider a binary string (a symmetric Bernoulli sequence) of length . For a positive integer , we exactly enumerate, in all? possible binary strings of length , the number of all runs of 1s of length (equal, at least)...Consider a binary string (a symmetric Bernoulli sequence) of length . For a positive integer , we exactly enumerate, in all? possible binary strings of length , the number of all runs of 1s of length (equal, at least)? and the number of 1s in all runs of 1s of length at least . To solve these counting problems, we use probability theory and we obtain simple and easy to compute explicit formulae as well as recursive schemes, for these potential useful in engineering numbers.展开更多
文摘In renewal theory, the Inspection Paradox refers to the fact that an interarrival period in a renewal process which contains a fixed inspection time tends to be longer than one for the corresponding uninspected process. We focus on the paradox for Bernoulli trials. Probability distributions and moments for the lengths of the interarrival periods are derived for the inspected process, and we compare them to those for the uninspected case.
文摘In several instances of statistical practice, it is not uncommon to use the same data for both model selection and inference, without taking account of the variability induced by model selection step. This is usually referred to as post-model selection inference. The shortcomings of such practice are widely recognized, finding a general solution is extremely challenging. We propose a model averaging alternative consisting on taking into account model selection probability and the like-lihood in assigning the weights. The approach is applied to Bernoulli trials and outperforms Akaike weights model averaging and post-model selection estimators.
文摘Consider performing a sequence of Bernoulli trials (each resulting in either a success, denoted S, or a failure F, with a probability of p and q := 1 - p respectively) until one of m specific strings (or patterns) of consecutive outcomes is generated. This can be seen as a game where m players select one such pattern each and the one whose pattern occurs first wins. We present symbolic formulas for the m probabilities of winning, and for the mean number of trials and the corresponding standard deviation to complete this game. Several numerical examples are presented, including a search for optimal strategy.
基金国家重点基础研究发展规划(973)(the National Grand Fundamental Research 973 Program of China under Grant No.2004CCA02500)国家自然科学基金(the National Natural Science Foundation of China under Grant No.60572015)湖北省教育厅优秀中青年人才项目(No.Q200726003)。
文摘A probability forecast method of earthquake magnitude, based on the earthquake frequency magnitude relation and the model of Bernoulli′s random independent trial, is applied to the earthquake risk assessment of seismic zones in China's Mainland before A.D.2005 in the paper. The forecasting results indicate that the probabilities of earthquake occurrence with magnitude 5 in seismic zones before 2005 are estimated to be over 0.7 in common and 0.8 in most zones; and from 0.5 to 0.7 with M =6; the maximum probability of earthquake occurrence with magnitude 7 is estimated at 0.858, which is also expected in Shanxi seismic zone. In west China's Mainland, earthquakes with magnitude 6 are expected to occur in most seismic zones with high probability (over 0.9 in general) ; the relatively high probabilities of earthquake occurrence (more than 0.7) with magnitude 7 are expected in the seismic zones surrounding the Qinghai Tibet plateau and south Tianshan seismic zone. A discussion about the result confidence and the relationship between the estimated probability and the possible annual rate of earthquake occurrence is made in the last part of the paper.
文摘Consider a binary string (a symmetric Bernoulli sequence) of length . For a positive integer , we exactly enumerate, in all? possible binary strings of length , the number of all runs of 1s of length (equal, at least)? and the number of 1s in all runs of 1s of length at least . To solve these counting problems, we use probability theory and we obtain simple and easy to compute explicit formulae as well as recursive schemes, for these potential useful in engineering numbers.
