Posterior prediction: Consider a pilot study in which n! - 15 children enrolled in special education classes were randomly selected and tested dren for a certain type of learning disability. In the pilot study, yi- 2 chil tested positive for the disability. a) Using a uniform prior distribution, find the posterior distribution of θ, the fraction of students in special education classes who have the disabilitv. Find the posterior mean, mode and standard deviation of and plot the posterior density. Researchers would like to recruit students with the disability to partici- pate 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 PrỞ2-y2Yı = 2), the posterior predictive distribution of Y2, as follows: i. Discuss what assumptions are needed about the joint distribution of (Yİ.Y2) such that the following is true: 0 ii. Now plug in the forms for PrỞ2-Y210) and pⓞY1 2) in the above integral 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 1 - 2. d) The posterior mode and the MLE (maximum likelihood estimate; see Exercise 3.14) of θ, based on data from the pilot study, are both θ 2/15. Plot the distribution PrỞ2 Y2|θ 0), and find the mean and standard deviation of Y2 given θ compare these results to the plots and calculations in c) and discuss any differences. Which distribution for Y2 would you use to make predictions, and why?