[25 points total] Suppose the likelihood function for some random variable Y is modeled as Normal(0, 1). Suppose the prior on O is 0 Normal(0, 1). a. [5 points] What is the posterior predictive distribution of the next observation, Ynew, we would draw from the same population given that we observed Y = 2? Use to plot the density of Y new | Y = 2. b. [10 points] Consider a Monte Carlo estimator of the mean of the posterior predictive distribution from (a), M Ynew YO M new j=1 where the new are i.i.d. draws from P(Y new |Y = 2). Write code to compute Y new for simulated Monte Carlo samples of size M = 10,..., 2000 in increments of 200. Plot these estimates against M and also plot the true mean of the posterior predictive distribution in the same figure for comparison. c. [10 points] Now compute Monte Carlo estimates of the probability that Ynew 0 for Monte Carlo samples of size M = 10,...,2000 in increments of 200. Make a plot of these estimates by M and plot the posterior predictive probability that Y new