## MCS 494: Review for the Midterm Exam

The questions below are meant to stimulate the preparation of the exam. They cover some of the most important but not all topics of chapters 1 to 8. You must review all course materials : textbook, homework exercises, and lecture notes (also consult the course web site). A scientific calculator may be used.

1. Consider the quality loss function L(X) = 48(X-115)^2. Use this function to evaluate two production processes with mean mu and standard deviation sigma: A: (mu,sigma) = (115,5) and B: (mu,sigma) = (114,2).
Which production process do you prefer: A or B? Justify your answer.

2. Assuming people are born on any given day in the week with equal probability, what is the chance that two or more students in the class (we are with 15) are born on the same day of the week?

3. Write pseudo code to simulate the traffic at a toll booth on a highway. The parameters are the number of toll booths, mean and standard deviation of processing time at the booth and of the arrival times of cars.

4. Find the z-transform of the signal 1,2,3,2,1,2,3,2,1,3,2,1,...

5. Solve the recursion x(k) = x(k-1) + x(k-2).

6. What is aliasing?

7. Show that the Discrete Fourier Transform turns convolutions in coordinate-wise products. Formally, we write
```         x * y ->  X.Y,  with X and Y the DFT of x and y.
```
( Hint: translate the statement for frames of four elements and show the property for these frames.)

8. Formulate the major steps in designing a filter. State the major theorem that is applied here.

9. Minimize x+3y subject to x ≥ 2, y ≥ 1, x+2y ≤ 8 and x+y ≤ 6. Graph the feasibility region.

10. Consider the data (x,y) given in the table
```             x | -2 | -1 |  1 | 2 | 3
---+----+----+----+---+---
y |  2 |  1 | -1 | 0 | 2
```
Formulate the normal equations you need to solve to find the best fitting quadratic y = a x^2 + bx + c.

11. Suppose you have the choice between receiving \$1,000,000 now, or 30 installments of \$60,000 each year. Figure out the critical value of the discount rate which makes both options equally attractive and indicate which option is best when the discount rate is lower or higher than this critical value.

12. Formulate a concept of total wealth of a nation. What is the effect of taxes on this total wealth?

13. What is elasticity of demand?

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

This list of questions is by no means exhaustive, but illustrates the type of questions on the exam: short definitions, derivation of a result, a calculation, or a general open-ended question.