Answer Happy

A galaxy cluster is an astronomical structure that contains anywhere from hun- dreds to thousands of galaxies bound together by gravity. Measurements of log-intensity (log number of photons) from small patches of the sky can be used to determine whether or not a galaxy cluster exists at a certain location. Galaxy clusters will emit a log-intensity that follows a normal distribution with mean 10 units and standard deviation 3 units. A log- intensity measurement from a part of the sky with no galaxy cluster present is distributed normally with mean 1 unit and standard deviation 1 unit. While there is no way of actually knowing if a galaxy cluster is located in a particular patch, an observation will be considered as having a galaxy cluster if the measured log-intensity has less than 0.02 probability of having come from a patch without a galaxy cluster. It is thought that the probability of a measurement actually being from a galaxy cluster is 0.01. i. Calculate the probability that an observation is considered as having a galaxy cluster. ii. Calculate the probability that an observation with log-intensity measurement greater than 3 units is from a galaxy cluster.