1. Acceptance/rejection methods. 

        1.a. Give an algorithm to generate a normal random  
             deviate using the acceptance/rejection method
             with the double exponential density as the 
             majorizing density.  After you have obtained the  
             acceptance/rejection test, try to simplify it. 

        1.b. Write a program to generate bivariate normal deviates 
             with mean $(0,0)$, variance $(1,1)$, and correlation $\rho$. 
             Use a bivariate product double exponential density as the 
             majoring density.  Now set $\rho=0.5$ and generate a sample 
             of 1,000 bivariate normals.  Compare the sample statistics 
             with the parameters of the simulated distribution. 
 
        1.c  What would be the problem with using a normal density 
             to make a majorizing function for the double exponential
             distribution (or using a half-normal for an exponential)? 

        1.d. Let $T$ be the number of passes through the three steps 
             until the desired variate is delivered.  Determine the 
             mean and variance of $T$ (in terms of $p_X$ and $g_Y$).  

        1.e. Now consider a modification of the rejection method in 
             which steps 1 and 2 are reversed, and the branch in step 3 is 
             back to the new step 2, that is: 

        (1)  Generate $u$ from a uniform (0,1) distribution. 
        (2)  Generate $y$ from the distribution with density function $g_Y$. 
        (3)  If $u \leq p_X(y)/cg_Y(y)$, then take $y$ as the desired 
             realization;  
             otherwise return to step (2). 

             Is this a better method? Let $Q$ be the number of passes  
             through these three steps until the desired variate is
             delivered.  Determine the mean and variance of $Q$.

2.  Use the Metropolis-Hastings algorithm to generate a sample of
    standard normal random variables.  Use as the candidate generating
    density, $g(x|y)$, a normal density in $x$ with mean $y$.  
    Experiment with different burn-in periods and different starting 
    values.  Plot the sequences generated. 

3.  Consider the trivariate normal distribution used as the example 
    in class.

      3.a.  Use the Gibbs method to generate and plot 1,000 
            realizations of $X_1$ (including any burn-in). Explain any 
            choices you make in how to proceed with the method. 
      3.b.  Use the hit-and-run method to generate and plot 1,000 
            realizations of $X_1$ (including any burn-in). Explain any 
            choices you make in how to proceed with the method.