Stat 5101 Final ExamDecember 20, 1999

Name Student ID Secret Code (if you want your final grade posted).

The exam is closed book. You may two $8 \frac{1}{2}$by 11 sheets of paper with formulas (or anything else) on it, but no other notes. You may use a calculator, but shouldn't need to. Put all of your work on this test form (use the back if necessary). Show your work or give an explanation of your answer. No credit for numbers with no indication of where they came from.

The points for the questions total to 100. There are pages and 8 problems. A normal distribution table is on a separate sheet of paper.

1. [25 pts.] Suppose X and Y are independent and identically distributed normal random variables. Show that
   $$\begin{align*}U & = X + Y \\ V & = X - Y \end{align*}$$
   are independent random variables. Be careful to clearly explain your reasoning.

2. [30 pts.] Suppose X and Z are independent random variables and
   $$\begin{align*}X & \sim \text{Exp}(\lambda) \\ Z & \sim \mathcal{N}(0, \sigma^2) \end{align*}$$
   Suppose Y = X² + Z.
   1. Find the function of X that is the best predictor (BP) of Y, also called the best unbiased predictor (BUP).
   2. Find the function of X that is the best linear predictor (BLP) of Y, also called the best linear unbiased predictor (BLUP).

      Hint: You may find the following integral formula helpful
      

      $$\int_0^\infty x^n e^{- \lambda x} \, d x = \frac{\Gamma(n + 1)}{\lambda^{n + 1}} = \frac{n !}{\lambda^{n + 1}}$$
      
      

3. [15 pts.] Suppose X₁, $\ldots$, X_n are independent and identically distributed normal random variables with mean 100 and standard deviation 15. Let $X{\mkern -13.5 mu}\overline{\phantom{\text{X}}}_n$ be the usual sample mean, and find $P(X{\mkern -13.5 mu}\overline{\phantom{\text{X}}}_n < 105)$.

4. [25 pts.] Suppose X, Y, and Z are independent and identically distributed random variables with mean $\mu$ and variance $\sigma^2$. What is the variance matrix (what Lindgren calls the covariance matrix) of the random vector
   
   $$\boldsymbol{W} = \begin{pmatrix}X \\ X + Y \\ X + Y + Z \end{pmatrix}$$
   
   

5. [30 pts.] A pair of random variables X and Y have joint density
   
   $$f(x, y) = k \exp(- x^2 - y^2 - x y + y)$$
   
   
   where k is a constant (its value does not matter). Find the conditional distribution of Y given X(you may specify it in any way that completely defines it, for example, by giving the conditional density or by giving the name of the conditional distribution and the values of its parameters).

6. [25 pts.] Suppose X and Y are independent standard normal random variables. Find the joint density of the variables
   $$\begin{align*}U & = X \\ V & = Y / X \end{align*}$$
   (This transformation is undefined when X = 0, but that event occurs with probability zero and may be ignored.)

7. [25 pts.] Assume that the locations of dogwood trees in a forest form a homogeneous Poisson process with a rate of 5.7 dogwood trees per acre. Let X be the number of dogwood trees in a region having an area of 20 acres. Approximate P(X > 100) using a normal approximation with continuity correction.

8. [25 pts.] A strictly positive random variable X has the probability density function f defined by
   
   $$f(x) = \frac{2}{(1 + x)^3}, \qquad x > 0.$$
   
   
   1. Find the cumulative distribution function (c. d. f.) of X.
   2. Find the median of X.