Statistics 610
Homework #7 
Assigned Wednesday April 28, 2010
Due in class Thursday May 6, 1210



1. Two count variables X and Y are independent, with Geometric(alpha)
   and Geometric(beta) distributions, respectively.  Recall, the p.m.f.
   of a Geometric(theta) variable Z is

   f(z;theta) = theta (1-theta)^{z-1}   for z = 1, 2, 3, ...

   and for theta in (0,1).  We wish to test the null hypothesis 
   H_0: alpha = beta.  Identify a sufficient statistic for the common
   value on H_0, and derive an exact test.  Compute the p-value
   when we observe x=3 and y=10.


2. Consider the two sample (unpaired) comparison in which we have
   mutually independent observations X_1, X_2, ..., X_m from one
   population, with c.d.f. F(x), and Y_1, Y_2, ..., Y_n from a
   second population, with c.d.f. G(x).  Suppose that both populations
   are continuous, and thus there are no ties in the data.
   To test the null hypothesis that F=G (for all x), we will consider the 
   Wilcoxon rank sum statistic

       W = sum_{i in sample 1} R_i     -  m(m+1)/2

   where R_i is the rank of observation i taken relative to the
   combined sample of X's and Y's.  Another statistic is

       S = sum_{i=1}^m sum_{j=1}^n 1[ X_i > Y_j ]

   Claim: W=S.  
   Prove the claim, at least in the special case that m =1, 2, or 3.

   Another procedure is to convert the original data to ranks, as in
   the construction of W, but instead to compute the two-sample t
   statistic (T) using the ranks as input.  Show that the p-value you would 
   get for either T or W is the same, when based on the permutation
   distribution.

3. a. Let Z ~ Beta(a,b) and Y|Z=z ~ Binomial[ n, z ].  Determine the
      marginal distribution of Y.  Determine the conditional distribution
      of Z given Y=y.
   b. Let Z ~ Gamma(a,b) and Y|Z=z ~ Poisson[ z ].  Determine the
      marginal distribution of Y.  Determine the conditional distribution
      of Z given Y=y.
   c. Let Z ~ Normal(a,b) and Y|Z=z ~ Normal[z, sigma2]. Determine
      the marginal distribution of Y.  Determine the conditional distribution
      of Z given Y=y.
   d. Let Z ~ Gamma(a,b) and Y|Z=z ~ Normal[0, z]. Determine
      the marginal distribution of Y.  Determine the conditional distribution
      of Z given Y=y.