A rancher sold 25 hogs and 60 sheep to Mr. Kay for \(\$ 3450 .\) At the same prices, he sold 35 hogs and 50 sheep to Mr. Bea for \(\$ 3300\). Find the price of each hog and sheep.

Suppose that two costume companies each make clown, skeleton, and space costumes. They all sell for the same amount and use the same machinery and workmanship. Furthermore, the market for costumes is fixed; a certain given number of total costumes will be sold this Halloween. But each company has its own individual styles which affect how the costumes sell. On the basis of past experience, the following matrix has been inferred. It indicates, for example, that if both make clown outfits, then for every 20 that are sold, company I will lose 2 sales to company II. Similarly, if company I makes clown outfits and company II makes space suits, then for each 20 sold, company I will sell 4 more than company II.

Consider a buyer who at the beginning of each month decides whether to buy brand 1 or brand 2 that month. Each month the buyer will select either brand 1 or 2 . The selection he makes at the beginning of any one month depends at most on the selection he made in the immediately preceding month and not on any other previous selections. Let \(\mathrm{p}_{\mathrm{ij}}\) denote the probability that at the beginning of any month the buyer selects brand \(j\) given that he bought brand i the preceding month. Suppose we know that \(p_{11}=3 / 4, p_{12}=1 / 4\), \(\mathrm{p}_{21}=1 / 2\) and $\mathrm{p}_{22}=1 / 2$ 1) First draw the transition diagram. 2) Assuming that the first month under consideration is January, determine the distribution of probabilities for April.

The technique used to estimate a relationship of the form \(\mathrm{Y}=\beta_{0}+\beta \mathrm{X}\) is known as simple regression. Generalize this technique to estimate a relationship of the form, $$ \mathrm{Y}=\beta_{0}+\beta_{1} \mathrm{X}_{1}+\beta_{2} \mathrm{X}_{2}+\ldots+\beta_{\mathrm{K}} \mathrm{X}_{\mathrm{K}} $$ where \(\mathrm{K}\) is some positive integer. This new technique is known as multiple regression.

1) Show that the standard deviation of \(\mathrm{X}\) and \(\mathrm{Y}\) corresponds to vector addition (according to the parallelogram law). 2) Solve the following problems: (a) Suppose that 3 resistors are placed in series. The standard deviations of the given resistances are 20,20 and 10 ohms, respectively. What is the standard deviation of the resistance of the series combination? (b) Gears \(\mathrm{A}, \mathrm{B}\) and \(\mathrm{C}^{\prime}\) s widths have standard deviations of \(0.001,0.004\), and \(.002\) in., respectively. If these are assembled side by side on a single shaft what is the standard deviation of this total width?

A highly specialized industry builds one device each month. The total monthly demand is a random variable with the following distribution. $$ \begin{array}{lllll} \text { Demand } & 0 & 1 & 2 & 3 \\ \mathrm{P}(\mathrm{D}) & 1 / 9 & 6 / 9 & 1 / 9 & 1 / 9 \end{array} $$ When the inventory level reaches 3, production is stopped until the inventory drops to \(2 .\) Let the states of the system be the inventory level. The transition matrix is found to be Assuming the industry starts with zero inventory find the transition matrix as \(\mathrm{n} \rightarrow \infty\).