A generalization of the hierarchy in Exercise 4.34 is described by D. G. Morrison (1978), who gives a model for forced binary choices. A forced binary choice occurs when a person is forced to choose between two alternatives, as in a taste test. It may be that a person cannot actually discriminate between the two choices (can you tell Coke from Pepsi?), but the setup of the experiment is such that a choice must be made. Therefore, there is a confounding between discriminating correctly and guessing correctly. Morrison modeled this by defining the following parameters:
p = probability that a person can actually discriminate,
c = probability that a person discriminates correctly.
Then

where 1/2(1 - p) is the probability that a person guesses correctly. We now run the experiment and observe X1,..., Xn ~ Bernoulli(c), so

However, it is probably the case that p is not constant from person to person, so p is allowed to vary according to a beta distribution,
P ~ beta(a, b).
(a) Show that the distribution of EXi is beta-binomial.
(b) Find the mean and variance of EXi.

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c=p+(1- p) =1+ p),