Problem 1:

Corky Miller, baseball player, has two prior distributions for, the probability of Corky getting a hit (a success) during an aim at bat (a trial): Beta(1, 1) and Beta(18, 56). Let's also consider a third prior distribution Beta(1/2, 1/2). Note that Beta(1/2, 1/2) is a "Jeffrey's prior", Beta(1, 1) is a uniform prior, and Beta(18, 56) is an informative prior. If Corky Miller gets one hit in his first 10 at bats of a season (1 successes in 10 trials), compare the following between the three different prior distributions:

The mean of the prior distribution for(use a formula, not simulation)

The mode of the prior distribution for(use a formula, not simulation)

The mean of the posterior distribution for(use a formula, not simulation)

The mode of the posterior distribution for(use a formula, not simulation)

A 80% equal tail posterior probability interval for(use the appropriate quantile function from software)

Please present these comparisons in an organized table.

Then please answer the following question: "Which of the prior distributions for(Jeffrey's, Uniform, Informative) would be the most sensible if you were gambling with friends and would win $10,000 if the mode of the posterior distribution you chose was closest to the batting average at the end of the season (after many new trials)?"

Problem 2

You are studying whether Gene A is a promoter of Gene B. You conduct a simple experiment, where you measure the number of times (counts) you see gene B over a standard interval of time. If gene A is a promoter of gene B then you expect to see 10 counts of gene B over the time interval. If gene A is not a promoter of gene B, then you expect to see 3 counts of gene B. You are unsure whether it is more likely that gene A is a promoter of gene B or that it is not. During the experiment you observe 7 counts of gene B. What is the posterior probability that gene A is a promoter of gene B? What is the marginal probability of the observed data?