A physical education student is studying the effect of a simple training program on basketball free-throw shooting. She recruits a random sample of 100 college students to participate in her study. Each student first shoots 100 free-throws to establish a baseline success probability. Each student then takes 50 practice shots each day for a month. At the end of that time, each student takes 100 shots for a final assessment. Let Yi denote the ith student's difference between his/her accuracy rate (shots made/100) after the program and his/her accuracy rate before the program, where "the program" is taking 50 practice shots each day for a month. Assume that each Yi|iidN(,2), where is an unknown parameter and 2 is a known number. Give two prior distributions for , explaining each choice in a sentence or two: 1.a diffuse (vague) prior 2. a subjective prior based on your knowledge of free throw shooting Find a coin that you can use for this exercise. Assume that the number of heads that come up when your coin is flipped n times is Bin(n, ), where is the probability that your coin lands heads up when flipped. Assume that before flipping it your prior distribution for is Beta(4,4). 1.Plot the prior density function for in R. 2. Flip the coin one time, clearly report the outcome, and find the posterior distribution for conditional on the outcome of the flip, carefully showing your derivation for full points. Plot the posterior density function in R. 3. Now assume that the prior distribution for is the posterior distribution from part (b). Flip the coin again and report the new outcome. Find the the posterior of given the new coin flip and plot the posterior density function. 4. Assuming the original Beta(4,4) prior distribution for , find the posterior distribution of conditional on the outcome of both coin flips. Compare this posterior distribution to the one you obtained in part (c).