# Solutions to Midterm Exam Mon 14 Oct 2002

1. Suppose the quality loss function of a manufactoring process is L(X) = 20(X - 12)^2.
What is the expected loss if the production methods yield a mean of 11 and a standard deviation of 2?

Solution:

The formula for the expected loss is E[L(X,theta)] = k*sigma^2 + k*(mu-theta)^2 when L(X,theta) = k*(X-theta)^2 is the quality loss function.

Applied to this problem, the target theta is 12, the mean is 11, and the standard deviation sigma is 2. So the expected loss is 10*(2^2 + (11-12)^2) = 100.

2. Find the z-transform of the signal 1,2,1,2,1,2,...

Solution:

Recall the series

```     1                              1
----- = 1 + z + z^2 + ...  =>  ----- = 1 + 1/z + 1/z^2 + ...
1 - z                          1-1/z
```
which leads to the z-transform of the signal 1,1,1,1,... To insert one zero between every pair of ones, we substitute z by z^2, and we obtain z^2/(z^2 - 1) as the z-transform of the signal 1,0,1,0,... After multiplication with 2/z, we obtain the z-transform of the signal 0,2,0,2,... The addition of these two z-transforms
```     z^2             2 z^2     z^2 + 2 z
------- + z^(-1) ------- = -----------
z^2 - 1          z^2 - 1     z^2 - 1
```
yields the z-transform of the signal 1,2,1,2,...

3. What is the most important property of a linear, causal, and time-invariant filter?

Solution:

If we know the output of the filter when giving it the unit impulse on input, then we can determine the output for any input. In formulas, let h_k, k=0,1,.., infinity, be the inpulse response, then

```               infinity
--------
\          -k
H(z) =  >     h  z
/       k
--------
k = 0
```
is the transfer function of the filter. In the time domain, for periodic input u, we have that the output y is y = h*u, with * the circular convolution operator.

4. Solve the following problem:
```            minimize  x1 + 2 x2
subject to
{ 6 x1 +   x_2 ≥ 6
{   x1 +   x_2 ≥ 3
{ 2 x1 + 5 x_2 ≥ 10
```

Solution:

Graphing the three constraints and drawing rays parallel to the objective function x1 + 2*x2, we see that the minimum occurs at the intersection of the second and third line. Solving this linear system, we obtain (x1,x2) = (5/3,4/3) with value 13/3 of the objective function.

As an alternative to the graphical solution, we could solve three linear systems and compare the values of the intersection points at the objective.

Another method (recommended for larger problems) is to set up the tableau and execute the simplex algorithm.

5. Compute the present value of \$1,000 five years from now, using a discount rate of 5%.

Solution:

We apply the inverse continuous compounding rule:

```            -0.05 x 5
\$1,000 e          = \$778.80.
```

6. Formulate the relation between elasticity of demand and the revenue of the producer.
Why is this relation so important?

Solution: The formulas for elasticity e and revenue R are

```          D'(p)
e = - -----    and R = p D(p)
D(p)
```
where p is the unit price and D(p) the demand function. A simple derivation of revenue with respect to price
```   dR                        (     D'(p))
-- = D(p) + p D'(p) = D(p)( 1+p -----) = D(p)(1-e)
dp                        (     D(p) )
```
shows the relation between change in revenue due to price increase and the elasticity of demand.

The importance of this relation follows. If e < 1, then 1-e > 0 and the revenue will go up when the price increases. If e > 1, then 1-e < 0 and the revenue will go down when the price increases. So elasticity determines whether the producer can increase the price and increase the revenue at the same time.

7. Write one paragraph on you learned from an unrelated project of another team.

Solution:

There are several good answers to this question, here are a few suggestions:

• We learned about the chances and expected income of a lottery, on how it depends on the size of the rewards and the number of participants.
• We learned about gathering data, either from a real shop, or through simulation, to study servicing requests.
• We learned about the image processing capabilities of MATLAB.

Expanding a bit on these suggestions makes a good answer.