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Probability : Mean, Distribution, Density and Exponential Distribution

On a particular examination paper, a pupil whose mean score is {see attachment} will actually score X, where X has a {see attachment} distribution. Over the pupils taking the test, {see attachment} is assumed to have a {see attachment} distribution. Find the density of the test score of a randomly chosen pupil.
Given that Fred has scored x, what is the distribution of his underlying mean score?

Suppose {Xi : i =1,2.....n} are independent, with Xi having the Exponential Distribution...
*Please see attachment for symbols and equations


Solution Summary

Density of data and an exponential distribution is investigated. The solution is detailed and well presented. The response received a rating of "5" from the student who originally posted the question.