Two political scientists wish to forecast the proportion of votes that a certain U.S. senator will earn in her upcoming reelection contest. The first political scientist assigns a Beta(3, 2) prior to p = the true support rate for the senator, while the second assigns a Beta(6, 6) prior.
a. Determine the expectations of both prior distributions.
b. Which political scientist appears to feel more sure about this prior belief? How can you tell?
c. Determine both Bayes estimates in this scenario, assuming that y out of n randomly selected voters indicate they will vote to reelect the senator.
d. For what survey size n are the two Bayes estimates guaranteed to be within .005 of each other, no matter the value of y?