The following discussion provides an alternative approach to determine a prior distribution for p. Earlier it was mentioned that the larger the values of the parameters α and β, the smaller the spread in the distribution. You can verify this by computing the variance of a beta random variable for small and large values of α and β. The mean of the beta distribution is α/(α + β), so fix E(p) = α/(α + β), equal to a desired value, but set a and b higher or lower depending on how much variability in p you are willing to allow. For example, perhaps you strongly believe that the prior estimate of the proportion of brown or orange M&M’s is E(p) = 0.33 and you are fairly certain that p does not vary too much. Then you might choose a = 50 and b = 100. However, if you are not too sure about the variability in the values of π, then you might select a = 1 and b = 2. Based on this approach, do the following:
a. Determine some reasonable values for α and β for the skeptic, the open-minded individual, and the believer in psychic abilities.
b. Assume that in a ganzfeld experiment, 8 hits are observed in 40 sessions. For each type of individual, plot the prior and posterior distributions on the same graph. Compare the changes in the characteristics (mean and variance) of the prior and posterior distributions for the three types of individuals. For which type of person did the posterior estimate of π change the most from the prior estimate of π?