[3] Let X be a Poisson (2) random variable representing the number of October surprises across all Senate races for randomly chosen October's in election years. The Apple party contended there were many instances and it was a big problem; the Orange party dismissed that point of view, claiming it was not a big issue over the ~33 races, certainly less than 10. Using a random sample of 5 election years, the Apple party estimated with the maximum likelihood estimate x. Let a random sample of data be {18, 13, 16, 11, 12) and x = 14 = 1, for the number of October surprises. But a political scientist wants to estimate for the Orange party using a Bayesian approach. The scientist played it honestly at first and cross-referenced candidate claims and social media positions to determine a gamma prior distribution for A, f^(^) = -2--B, gamma (, ), where a and B are known as the shape and scale, respectively, () for the gamma distribution. The scientist chose gamma (a = 10, p = 2) to represent the prior position. The posterior distribution for A is then a gamma(s + a, n+), where s = =1 Xi. [a] Assuming a squared error loss function for the Bayesian estimate, what is the political scientist's Bayesian estimate ? [b] Then partisanship set in for the political scientist who had Orange party leanings. So the scientist then maintained the prior mean, but chose a gamma (a = 50, p = 10). The scientist assumed the slight-of-hand giving more weight to the prior would go unnoticed. Under the second prior, again with squared error loss, what is the Bayesian estimate for ?