The math symbols in the following are written in TeX.

1.      Consider the problem of estimating $\mu$ and $\sigma$ (the mean
        and standard deviation) in a normal distribution.  For
        estimators in a sample of size $n$, we will use the sample mean,
        $\bar{x}_n$, and the sample standard deviation, $s_n$.
        Assume that
    \[
         {\rm MSE}(\bar{x}_n) = O(n^{\alpha})
    \]
        and 
    \[
         {\rm MSE}(s_n) = O(n^{\beta}).
    \]
        Perform a Monte Carlo experiment to estimate $\alpha$ and
        $\beta$.  Plot your data on log-log axes, and use least squares
        to estimate $\alpha$ and $\beta$.  Now, derive the exact values
        for $\alpha$ and $\beta$ and compare with your estimates.

2.      Use Monte Carlo to study the performance of the histogram density 
        estimator $\widehat{f}$ using normal data.  
        Generate samples of size 500 from a
        $N(0,1)$ distribution.  Use a Monte Carlo sample size of 100.

     a.  Choose three different bin sizes.  Tell how you chose them.
     b.  For each bin size, estimate the variance of $\widehat{f}(0)$.
     c.  For each bin size, compute the average IMSE.

3.   The basic problem in density estimation as follows:
     Given a random sample $x_1, x_2, \ldots, x_n$ from an unknown 
     population, estimate $Pr(X\in S)$ for a random variable $X$ 
     from that population, or estimate $p(x_0)$, where $p$ is the 
     probability density, or estimate the function $p$ or $P$.

           Suppose, instead, the problem is to generate random numbers
           from the unknown population.  

           a.     Describe how you might do this.  There are several
                  possibilities that you might want to explore. 
                  Also, you should
                  say something about higher dimensions.

           b.     Suppose we have a random sample as follows: 
                 -1.8, -1.2, -.9, -.3, -.1, .1, .2, .4, .7, 1.0, 1.3, 1.9
                 Generate a random sample of size 5 from the population
                 that yielded this sample. Again, you must
                  describe your procedure in detail.  It would 
                  be good to carry out the test on the computer,
                  but a computer program by itself is not a satisfactory
                  solution. 
 
4. The $L_p$ error of bona fide density estimators.

    a. Show that the $L_1$ error is less than or equal to 2.

    b. By an example, show that the $L_2$ error has no bound.