Temple University
Department of Economics

Economics 8009
Midterm - Fall 2008
$1 for a red, there are 24

$2 for a club, there are 12

$7 for a spade from 2 to 9, there are 8

$10 for 10, jack, queen, king of spades, there are 4

Note that we have accounted for the 48 cards remaining in the deck.
A. Prob distribution for payoff

B. Mean of x

C. Variance of x

2. Show (x1+x2)/2 is an unbiased estimator of the the rate parameter in a Poisson process.
A de minimus answer would be to argue that the expected value of a Poisson random variable is equal to l.  Therefore, the expected value of x1 is l and the expected value of x2 is also l.  The expected value of their sum is 2l, hence the expected value of their average is just l.

In a more complete answer you would start by proving that the expected value of a Poisson random variable is l, then pick up where the above paragraph begins.
3. An industrial plant has the null hypothesis that m = 85. 

If output is too great then you end up with too much inventory.  If you produce too little then you cannot meet demand.  Therefore this is a two tail test. Use a Student's  t-test. First get the critical values.

Reject the null hypothesis
4. Quadratic Form

4.A. Find the roots
Determinantal equation

4.B. Find the latent vectors

We get the two equations

To make sure that we get a non-trivial, unique solution we impose the restriction that the latent vectors have unit length. 

=0
We can rewrite the restriction as

Upon substitution

And now solve for the two unkowns

For the other latent vector we get the two equations

C. Pre and post multiply

D. Find L

5. Regression problem
A.The sample size

B. Regression coefficients

First we need the inverse.  Expand along the first column to find the determinant
Detx =

The matrix of cofactors is

C. Variance of the error term

D. Coefficient Covariance

E. Ho: a2=0

F. Linear Combinarion: Same critical values as part E