Posterior prediction: Consider a pilot study in which n1 = 15 childrenenrolled in special education classes were randomly selected and tested for a certain type of learning disability. In the pilot study, y1 = 2 children tested positive for the disability.a) Using a uniform prior distribution, find the posterior distribution of _, the fraction of students in special education classes who have the disability. Find the posterior mean, mode and standard deviation of _, and plot the posterior density.Researchers would like to recruit students with the disability to participate in a long-term study, but first they need to make sure they can recruit enough students. Let n2 = 278 be the number of children in special edu- cation classes in this particular school district, and let Y2 be the number of students with the disability.b) Find Pr(Y2 = y2|Y1 = 2), the posterior predictive distribution of Y2, as follows:Discuss what assumptions are needed about the joint distribution of (Y1,Y2) such that the following is true: 1 0Now plug in the forms for Pr(Y2 = y2|_) and p(_|Y1 = 2) in the above integral.Pr(Y2 = y2|Y1 = 2) =Pr(Y2 = y2|_)p(_|Y1 = 2) d_ .iii. Figure out what the above integral must be by using the calculus result discussed in Section 3.1.c) Plot the function Pr(Y2 = y2|Y1 = 2) as a function of y2. Obtain the mean and standard deviation of Y2, given Y1 = 2.
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