The due date for this theme is Friday, April 12, at noon.

Reading assignment: Section 4.8, 4.9

Background reading: Section 4.5

A Markov chain is a mathematical model with many varied applications. A Markov chain is a simple difference equation. In this theme, we will study difference equations and then some basic properties of Markov chains.

The space consists of discrete-time signals. Give two examples of elements of (different from those in the book). Is the set of such that is an even integer for all a subspace of ? What about the set of such that is an odd integer for all ? Explain.

What is a linear difference equation? Consider the second-order linear difference equation:

Suppose you know that this equation has a solution that is linear function of . Find such a solution. Find a basis for the set of solutions of the corresponding homogeneous equation. How do you know this set is a basis? That is, how do you know the vectors are linearly independent and span the set of solutions of the homogeneous equation? Use this to find a general solution for the equation above.

Consider again the homogeneous linear difference equation corresponding to the equation above. Write this homogeneous equation as a first-order difference equation.

What is a Markov chain? Be sure to define any words or phrases you use to answer this question. We are going to use the following situation to study Markov chains.

New Mexico State University has decided to embark on a fund-raising campaign to upgrade classrooms and educational technology for the benefit of its students. After reviewing donating habits, the university's foundation discovered the following information. Of the alumni who did not donate the previous year, only 10% will donate this year, and they will all give small gifts. Of those who donated small gifts last year, 20% will not donate this year and 10% will donate large gifts; the remainder will donate small gifts. Finally, 70% of the donors of large gifts last year will give large gifts this year, 20% will give small gifts and the rest will fail to donate.

Suppose that the donating habits from last year are given by the probability vector

in which the first entry is the proportion of people who do not donate, the second entry is the proportion of people who give small gifts, and the last entry the proportion giving large gifts. Assume that the donating habits of alumni form a Markov Chain. Find the donating habits (i.e., probability vectors) for the next three years and explain what these vectors mean. (Do not round your answers.)

What is a steady-state vector? What is the steady-state vector for the example above? What does this steady-state vector say about how the long-term donating habits of NMSU alumni will change from last year's donating pattern?