Apply Gershgorin's theorem to estimate the eigenvalues of the matrix $$ \begin{array}{rlrr} & \mid(5 / 2) & -(1 / 2) & 1 \\ \mathrm{~A}= & (3 / 2) & (1 / 2) & -1 \mid \\ & 0 & 0 & 0 \end{array} $$

Let \(p(x)=6+2 x^{2}-6 x^{4}+4 x^{5}-3 x^{6}+x^{8}\). Apply Cauchy's polynomial root theorem to find a circle of radius \(\mathrm{r}\) within which all the roots of \(\mathrm{p}(\mathrm{x})\) lie.

Which of the following matrices can be interpreted as perfect communications matrices? For those that are, find the two stage and three stage communication lines that are feedback. $\begin{array}{lllllllll} & \mathrm{A} & \mathrm{B} & \mathrm{C} & \mathrm{D} & & \mathrm{A} & \mathrm{B} & \mathrm{C} \\\ \mathrm{A} & 0 & 1 & 1 & 2 & \mathrm{~A} & 0 & 1 & 1 \\ \mathrm{~B} & 1 & 0 & 1 & 0 & \mathrm{~B} & 1 & 0 & 1 \\ \mathrm{C} & 1 & 1 & 0 & 1 & \mathrm{C} & 1 & 1 & 0 \\ \mathrm{D} & 2 & 0 & 1 & 0 & & & & \end{array}$

Consider the following two-person game. Player 1 has \(\$ 3.00\) and player 2 has \(\$ 2.00\). They flip a fair coin; if it is a head, player 1 pays player $2 \$ 1.00\( and if it is a tail, player 2 pays player \)1 \$ 1.00$. How long will it take for one of the players to go broke or win all the money?

There are three dairies in a community which supply all the milk consumed: Abbot's Dairy, Branch Dairy Products Company, and Calhoun's Milk Products, Inc. For simplicity, let's refer to them hereafter as $\mathrm{A}, \mathrm{B}\(, and \)\mathrm{C}$. Each of the dairies knows that consumers switch from dairy-to-dairy over time because of advertising, dissatisfaction with service, and other reasons. To further simplify the mathematics necessary, let's assume that no new customers enter and no old customers leave the market during this period. Consider now the following data on the flow of customers for each of the dairy companies as determined by their respective Operations Research Departments: Flow of CustomersGiven the above data as well as all the assumptions made, determine a) The one-step transition probabilities b) Interpret the rows and the columns of the matrix of transition probabilities

A camera store stocks a particular model camera that can be ordered weekly. Let \(D_{1}, D_{2}, \ldots\), represent the demand for this camera during the first week, second week, \(\ldots\), respectively. It is assumed that the \(D_{i}\) are independent and identically distributed random variables having a known probability distribution. Let \(\mathrm{X}_{0}\) represent the number of cameras on hand at the outset, \(\mathrm{X}_{1}\) the number of cameras on hand at the end of week one, \(\mathrm{X}_{2}\) the number of cameras on hand at the end of week two, and so on. Assume that \(\mathrm{X}_{0}=3 .\) On Saturday, night, the store places an order that is delivered in time for the opening of the store on Monday. The store uses the following \((\mathrm{s}, \mathrm{S})\) ordering policy. If the number of cameras on hand at the end of the week is less than \(\mathrm{s}=1\) (no cameras in stock), the store orders (up to) \(\mathrm{S}=3\). Otherwise, the store does not order (if there are any cameras in stock, no order is placed). It is assumed that sales are lost when demand exceeds the inventory on hand. Assuming further that \(\mathrm{X}_{\mathrm{t}}\) is the number of cameras in stock at the end of the \(\mathrm{t}^{\text {th }}\) week (before an order is received, and that each \(D_{t}\) has a Poisson distribution with parameter \(\lambda=1\), determine a) the one-step transition probabilities b) Given that there are two cameras left in stock at the end of a week, what is the probability that there will be three cameras in stock two weeks and four weeks later. c) Compute the expected time until the cameras are out of stock, assuming the process is started when there are three cameras available; i.e., the expected first passage time, \(\mu_{30}\), is to be obtained. d) Determine the long-run steady-state probabilities.