Please use python to solve this problem
Problem 2
A particle physics experiment is searching for dark matter by looking for a signal of dark matter particles bumping into nuclei and causing nuclei to recoil. They have a detector that is able to ”see” the signature of a nuclear recoil. They have also gone to great pains to limit sources of background that may cause spurious signals, e.g., cosmic rays, radioactivity in or near the detector, etc.
They take data for a whole year, but despite their best efforts they are not able to eliminate all backgrounds. Through all sorts of difficult studies they conclude that they should expect on average to see ?B = 1.5 background events that mimic the dark matter signature. They see N = 2 candidate dark matter events.
Obviously, they do not have any evidence for dark matter interactions in their detector, since for a mean of 1.5 background events, the 2 events that they have could easily be all background. However, one can still extract information from such an experiment.
Clearly their result is incompatible with Nature being such that on average ?S = 1000 dark matter events would have been seen, since they only see N = 2, and the average background level was only ?B = 1.5. On the other hand, the experiment would have been insensitive to a scenario where ?S = 0.01. So what they need to do is to report a “limit” on ?S, i.e., the largest possible value of ?S that the experiment is statistically sensitive too.
Conventionally, what is done is to find the highest possible value of ?S such that the experiment would have seen ? N events (i.e., 0, 1, or 2) with > 5% probability. This is the so-called the “95% confidence level upper limit on ?S”. Technically this method to quote a 95% limit is called “Classic Frequentist”. There are several other statistical analysis methods on the market.
In order to calculate the limit, find the ? ? ?B +?S such that p(0|?)+p(1|?)+p(2|?) = 0.05, where p(n|?) is the Poisson probability of having n counts for a mean ?. Since ?B is given with only one digit after the decimal point, there is no need to be more precise than that when quoting ?S
A real statistical analysis would be more complex because it would have to account for an uncertainty in ?B (nothing is ever perfectly known in physics). Eventually, the results must be reported in some units that other people can understand. So for example ?S must be scaled by the fact that in this particular experiment the efficiency for seeing a nuclear recoil is not 100%, and the efficiencyis not perfectly known either, etc.