Scientists are using a device to count neutrinos. Every time a neutrino is detected, the sensor...

 

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Scientists are using a device to count neutrinos. Every time a neutrino is detected, the sensor will record the time. Neutrinos are assumed to arrive independently with a constant likelihood of arrival over the course of the experiment. The experiment lasts for one year. (a) What would be a good model for the number of neutrinos counted in this one experiment? What do the parameter (s) mean? (b) Now the scientists want to be able to say how long they are expected to wait before the next neutrino is detected. What distribution would you recommend to be used and why? Interpret the parameter(s). (c) After some puzzling results, a scientist realizes that the tool for counting neutrinos does not always get triggered. It turns out that the transistor in the equipment requires a high voltage to trigger the count, but sometimes the trigger will happen also at lower voltages. Below 5 micro volts the transistor never triggers, then the probability of triggering increases monotonically until 5.5 micro volts, at which point the transistor always triggers. The scientist wants to use a probabilistic model for the level of voltage when there is uncer- tainty about whether the transistor triggers the count or not. What is a good distribution and what range would its parameter(s) take to match the described behaviour of the transistor? Hints: • It is possible to change the support of a random variable (i.e. the set of values that can be taken on by the random variable) by a suitable transformation of that random variable. • It may be helpful in answering this question to explore the shape of the densities for distributions we studied in class, for example, using the applets provided under the following link: http://homepage.stat.uiowa.edu/~mbognar/. Scientists are using a device to count neutrinos. Every time a neutrino is detected, the sensor will record the time. Neutrinos are assumed to arrive independently with a constant likelihood of arrival over the course of the experiment. The experiment lasts for one year. (a) What would be a good model for the number of neutrinos counted in this one experiment? What do the parameter (s) mean? (b) Now the scientists want to be able to say how long they are expected to wait before the next neutrino is detected. What distribution would you recommend to be used and why? Interpret the parameter(s). (c) After some puzzling results, a scientist realizes that the tool for counting neutrinos does not always get triggered. It turns out that the transistor in the equipment requires a high voltage to trigger the count, but sometimes the trigger will happen also at lower voltages. Below 5 micro volts the transistor never triggers, then the probability of triggering increases monotonically until 5.5 micro volts, at which point the transistor always triggers. The scientist wants to use a probabilistic model for the level of voltage when there is uncer- tainty about whether the transistor triggers the count or not. What is a good distribution and what range would its parameter(s) take to match the described behaviour of the transistor? Hints: • It is possible to change the support of a random variable (i.e. the set of values that can be taken on by the random variable) by a suitable transformation of that random variable. • It may be helpful in answering this question to explore the shape of the densities for distributions we studied in class, for example, using the applets provided under the following link: http://homepage.stat.uiowa.edu/~mbognar/.

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Answer Solution a This model is detected or arrived for Nautrinos reco... View the full answer