Problem 3 (25 points). It is believed that cross-fertilized plants produce taller offspring than self-fertilized plants. In order to obtain an estimate on the proportion of cross- fertilized plants that are taller, an experimenter observes a random sample of 15 pairs of plants that are exactly the same age. Each pair is grown in the same conditions, with some cross-fertilized and the others self-fertilized. Based on previous experience, the experimenter believes that the following are possible values of 0 and that the prior probability for each value of is (О). 0: 0.80, 0.82, 0.84, 0.86, 0.88, 0.90 (0) 0.13, 0.15, 0.22, 0.25, 0.15, 0.10 From the experiment, it is observed that in x = 13 of 15 pairs, cross-fertilized is taller. (a) Create a table with columns of the prior (0), likelihood of L(x|0) for different values of and for the given sample, prior times likelihood, and the posterior probability of 0. (b) Based on the posterior probabilities, what value of is the most likely? Also find E(x) based on the posterior probabilities. (c) Calculate the MLE of 0 and compare it with the Bayesian estimate.