Homework #3 for Stat 610

Due Thursday March 1, in class.

0. Let X and C be independent random variables.
   Show that P[ X > C | C=c ] = P[ X > c ].

1. From Wasserman,

   Chapter: 9
   Problems: 4; 6a,b,c; 

2. Suppose data X_1,...X_n constitute a random sample
   from a normal population with mean mu and standard
   deviation sigma.  Derive the maximum likelihood estimator
   of (mu,sigma).

3. Suppose data form independent pairs { (x_i, Y_i) i = 1,2,...,n  }
   where {x_i} are non-random constants and
   Y_i ~ Normal[ alpha + beta x_i, variance=sigma^2 ]
   Derive the MLE for (alpha,beta,sigma) and relate it
   to the least squares estimate of (alpha,beta).  Consider
   a matrix representation in which X is a n x 2 matrix with
   a first column of 1's and a second column holding x_i's,
   and Y is an n x 1 vector holding the Y_i's.  Express the estimate
   of (alpha,beta) as a solution of a linear system with
   these objects.

4. Identify MLE of (alpha,beta) in model that extends the
   regression model in problem 4 by allowing the variance
   of Y_i to be sigma^2_i.  Suppose that these variances are known.
   Similarly write the estimate as the solution of a certain
   linear system.
   
5. Zero-inflated Poisson:  Data X_1,...,X_n form a random
   sample of counts from the following  model

    f(x;theta) = (1/2) 1[x=0] + (1/2) exp( -theta ) theta^x /x!

   for theta>0 and x=0,1,2,...

   Write the likelihood and loglikelihood for theta.  
   Compute the MLE given the following realization, n=10:

              0 5 0 7 4 0 0 0 0 4

6. Mutant frequency problem: As discussed in class, on one 96 well
   plate we have plated cells at dilution m_a = 10^4 cells per well
   and on a second we have cells at dilution m_b = 10^5 cells per well.
   We observe S_a = 5 fertile wells on the first plate and S_b=50
   on the second plate. Compute the likelihood for the HPRT mutant
   frequency theta, and evaluate the MLE.