Arizona State University College of Liberal Arts and Sciences
   
 

Assignments for STP 598 (to be turned in by deadline)

Instructor: Ananda Majumdar

Arizona State University, Tempe

Some of the assignments are from the text book, and others are from the class notes given almost every class. Students are strongly encouraged to submit all the assignments. Electronic submissions are encouraged! Please send them to the instructor at the email address ananda at math dot asu dot edu.



Assignment 1 (due 2 Sep in class or by email by 7 PM {email submission is encouraged!})

Problems 1. Find the
(a) moment,
(b) variance,
(c) characteristic function
of a Uniform(a, b) random variable explicitly.
(d) If X follows Uniform(a, b) then show that Y = (X-a)/(b-a) has a Uniform(0, 1) distribution.

Problem 2. Find the
(a) moment,
(b) variance,
(c) characteristic function
of a Exponential(lamda) random variable explicitly.
(d) Show the memoryless property of the Exponential random varible X, i.e, that P(X > t+s | X> s) = P(X>t) for all s, t >0.

Problem 3. Using the characteristic function of Normal random variables show that if X ~ N(mu, sigma^2) then Z = (X-mu)/sigma has a Normal(0,1) distribution.

Problem 4. Find explcitly the natural conjugate prior for the Normal(theta, sigma^2) family where sigma^2 is a fixed number and theta is the parameter of interest.

Problem 5. Find explicitly the natural conjugate prior for lognormal (theta, sigma^2) family where sigma^2 is a fixed number and theta is the parameter of interest.

Problem 6: Find explicitly the natural conjugate prior for the Poisson(lambda) family.

Problem 7: From the notes given in class, state the posterior predictive distribution of a Beta-Binomial conjugate family, and display this distribution for a given n (say n = 5) and x=0, 1, 2, ..., n via bar diagram.



Assignment 2 (due 9 Sep in class or by email by 7 PM {email submission is encouraged!})

Problems 1: When X1, ..., Xn are iid (conditional on theta) N(theta, sigma^2) with sigma^2 fixed, and theta ~ Normal(mu, tau^2), find the posterior predictive distribution of Y given X1, ..., Xn (marginally, i.e there is no theta in the posterior predictive distribution).

Problem 2: With the posterior predictive distribution of Y given X1, ..., Xn, in the 1st problem, state a Bayesian estimate for Y, and find a 95% posterior predictive interval for Y.

Problem 3: Generate values of X1,.., Xn iid from Normal(theta, sigma^2) with values theta = 10 and sigma = 1 with n=50. Now assume that theta is unknown, sigma^2 is known, so from the Bayesian approach, assuming a Normal(mu, tau^2) prior for theta where mu = 0 and tau = 10, find explicitly the posterior distribution of theta given X1, X2,.., Xn.

Problem 4: Using the same set up as problem 3, draw graphs of the (i) likelihood of Xbar|theta=10, (ii) the prior of theta, and the (iii) posterior distribution of theta given X1, X2, .. Xn -- all in one picture. Give your comments.

Problem 5: Show that the posterior distribution of a parameter (in a one parameter family) given the data, is a function of the sufficient statistic, when it exists.

Problem 6: Find the Jeffrey's prior for theta when the likelihood under consideration is Binomial(n, theta).

Assignment 3 (due 12 Oct in class or by email by 7 PM {email submission is encouraged!})

Problems:
Chapter 3
3, 4

Chapter 5
3, 6



Assignment 4 (due 2 Nov in class)

Generate random samples Yij|mu_i, sigma^2 independent Normal(mu_i, sigma^2) i=1,2,3 and j=1,..,30 where mu_1=0, mu_2=-1, mu_3=2 and sigma^2=1. Assuming {Yij}i=1,2,3, j=1,..,30 is the data, and that the values of the parameters are unknown, let the joint prior of mu_1, mu_2, mu_3 and sigma^2 be marginally independent, with mu_i ~ Normal(mu_0, tau^2), and p(sigma^2) = 1/sigma^2 (the latter is an improper prior).


Problems:

a) Derive the Gibbs sampling procedure explicitly for this problem, assuming mu_0 and tau^2 are known values.
b) Using the Gibbs sampling procedure in (a), assume that mu_0 = 0 and tau^2=100. Now use these values of the hyper parameters, to generate from the posterior distribution of the parameter vector (mu_1, mu_2, mu_3, sigma^2), with the resulting chain a Markov chain that converges to the true posterior distribution of parameter vector (mu_1, mu_2, mu_3, sigma^2).
c) Check covergence of the above Markov chain, check autocorrelation, and finally, summarize the findings of the posterior distribution of the parameter vector (mu_1, mu_2, mu_3, sigma^2) stating explicitly how you did this (please attach any codes used).



Assignment 5 (Due 23 November 2009 Monday in the class)

Problems:
1. The n cross 1 data vector is Y. The design matrix is X, the regression coefficient vector is beta and the variance for Yi is sigma^2. Suppose Y|beta, sigma^2 has a multivariate Normal distribution with mean Xbeta, variance covariance matrix given by sigma^2 * I, where I is the identity matrix. Suppose we have the following prior in the model, The prior of pi(beta, sigma^2) = pi(beta) pi(sigma^2) where beta has a Multivariate Normal prior with mean 0 and variance covariance matrix V_beta, and sigma^2 has an Inverse Gamma(c, d) distribution. Find the Gibbs sampler for this problem, and state all steps clearly and logically.

2. The n cross 1 data vector is Y. The design matrix is X, the regression coefficient vector is beta and the variance for Yi is sigma^2, while the correlations Corr(Yi, Yj) = rho for all i not equal to j. Suppose, Y|beta, sigma^2, rho, has a multivariate Normal distribution with mean Xbeta , variance covariance matrix given by sigma^2 * R, where R is the correlation matrix of Y (conditional on the parameters). Suppose we have the following priors in the model, The prior of pi(beta, sigma^2, rho) = pi(beta) pi(sigma^2) pi(rho) where beta has a Multivariate Normal prior with mean 0 and variance covariance matrix V_beta, and sigma^2 has an Inverse Gamma(c, d) distribution where as rho has a Uniform(-1, 1) distribution. Find the Gibbs sampler for this problem, and state all steps clearly and logically.


Assignment 6

Problems: TBD