MAT9004 Assignment 3 Due at the beginning of your support class in week 12. 1. The temperature distribution in a metal plate ({(x, y) R2 : |x| 10, |y| 5}) is given by T (x, y) = x2 10x + 4y 2 + 9 (a) Sketch on the same plot the sets of points with temperature 0, 1, 1. [2] (b) From the point on the plate located at (1, 1), in what direction is the temperature increasing most rapidly? Decreasing most rapidly? [2] (c) Find the maximal and minimal values of temperature on the plate. [4] 2. The company specialising on mobile devices is about to release their new product Gal-Phone 8. From their experience of previous years it is possible to approximate the expected sales S of this product by S(x, y) = 11 3x + 2y, where x corresponds to its price and y 2 corresponds to resourses spent on advertising. Based on this prediction the company produces exactly S(x, y) devices. Then, the total profit is P (x, y) = xS(x, y) y 2 c S(x, y), where c corresponds to production costs of one Gal-Phone 8. Assuming that c = 1, help the company to find the best price-advertising strategy (x, y) to maximize the profit P (x, y) in the range {0 x, y 4}. [10] 3. A manufacturer has five seemingly identical computer terminals available for shipping. Unknown to her, two of the five are defective. A particular order calls for two of the terminals and is filled by randomly selecting two of the five that are available. (a) List the sample space for this experiment. [2] (b) Let A denote the event that the order is filled with two non-defective terminals. List the sample points in A. [2] (c) Find the probability of event A. [2] 4. A poker hand consists of five cards from a standard 52 card deck with four suits (spades, clubs, diamonds, hearts) and thirteen values in each suit (2, 3, , 10,Jack, Queen, King, Ace); the order of the cards in a hand is irrelevant. A straight consists of 5 cards of five consecutive ranks (Ace and 2 are not consecutive). A flush consists of 5 cards of the same suit. How many hands are there that are neither a straight nor a flush, and also do not include the ace of diamonds? [6] 5. An insurance company issues a one-year $1000 policy insuring against an occurrence A that historically happens with a probability 0.02. Administrative fees are $15 per policy and are not part of the companys profit. (a) How much should the company charge for the policy if it requires that the expected profit per policy be $50? [Hint: If C is the premium for the policy, the companys profit is C 15 if A does not occur and C 15 1000 if A does occur.] [2] (b) The company has sold the policy to three people. Let X be the number of the persons (out of 3) to whom A happens. Find the distribution of X. [3] (c) Take the premium C in (a). Find the distribution of the company's profit, the expectation and standard deviation. [5]