Stat 5101 Second Midterm ExamDecember 1, 1999

Name Student ID

The exam is closed book and closed 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 5 problems. A normal distribution table is on a separate sheet of paper.

1. [20 pts.] The random variables X and Y have joint p. d. f.
   
   $$f(x, y) = \tfrac{1}{2} (x + y) e^{- x - y}, \qquad x > 0,\ y > 0.$$
   
   
   Find the joint density of the random variables U and V defined by
   $$\begin{align*}U & = X + Y \\ V & = \frac{X}{X + Y} \end{align*}$$
   Indicate the ranges of the variables in your answer.

   
   

2. [20 pts.] Suppose $X \sim \text{Gam}(\alpha, \lambda)$. Find $E(\sqrt{X})$.

   
   

3. [20 pts.] Assume that the times when accidents occur on a particular stretch of Interstate 94 in the twin cities form a homogeneous Poisson process with a rate of 0.2 accidents per day. Find each of the following related distributions. You do not need to give any formulas, just the name of the distribution and its parameter or parameters. Explain each answer.
   1. What is the distribution of the number of accidents that occur in a one-week time period?

      
      

   2. What is the distribution of the amount of time that elapses between now and the time of the tenth accident (counting only accidents that occur in the future).

      
      

   3. What is the distribution of the number of weeks out of the next 20 weeks in which no accidents occur?

      
      

4. [20 pts.] Suppose X is the number of heads in 100 coin flips, and we want to know $P(X \ge 60)$? Approximate this probability using a normal approximation with continuity correction.

   
   

5. [20 pts.] Consider two random variables X and Y. The marginal distribution of X is $\text{Exp}(\lambda)$ and the conditional distribution of Y given X is $\text{Exp}(X)$, that is, exponential with parameter X.
   1. What is the p. d. f. of X? Indicate the range of the variable in your answer.

      
      

   2. What is the conditional p. d. f. of Y given X? Indicate the range of the variable in your answer.

      
      

   3. What is the joint p. d. f. of X and Y? Indicate the ranges of the variables in your answer.

      
      

   4. Find the marginal p. d. f. of Y. Indicate the range of the variable in your answer.