MAT9004 Assignment 3 School of Mathematics Monash University A/Prof Heiko Dietrich Prof Nick Wormald Due: 4pm Monday, 2 November 2020 (submission only online via Moodle; no cover sheet required) Present your solution so that the sequence of logical steps is clear, with succinct justfication for each step where it is not obvious. Note that marks will be deducted for solutions that do not show all the important logical steps and reasoning in reaching a conclusion. Answers should consist of complete sentences with a mixture of English words and mathematical symbols. Marks can also be deducted if solutions are much more convoluted than required. Handwriting must be legible and all sketches must be neat. It is your responsibility to verify that what you upload is a complete and legible scan of your work. Assignments received after the due date will receive a late penalty, so we encourage you not to plan to submit last minute. Extensions are only given in exceptional cases, please consult Moodle and the special consideration guidelines. This assignment has 60 marks and is worth 15% of the final unit mark. Question 3.1 (14 marks; Lecture 16) Your friend has 400m of wire and wants to build the frame of a rectangular block as shown here: They want to use all their wire and they want that the volume of the block is as big as possible; it remains to determine the side lengths a, b, and c of the block that achieve this. Being in Week 9 of MAT9004, you are by now an expert in 2-variable optimisation so you’ve offered your friend to help them out by doing the following: a) Explain in detail how this problem is the same as finding the global maximum of a 2-variable func- tion; you can arrange that this function has variables a and b, call if f(a, b). Write down a formula for f(a, b) and determine a suitable domain D such that the answer to your problem is the global maximum of f(a, b) on D. b) Determine all stationary points of f(a, b) on D. c) Determine the Hessian matrix of f(a, b) and classify each stationary point. d) Now do whatever is left to do in order to determine the global maximum of f(a, b) on D. e) What side lengths a, b, and c should your friend choose in order to get a block with maximal volume? What’s the maximal volume? Question 3.2 (10 marks, Lecture 16) Your friend has had enough with the wire and now they’re into moto-cross. They have a big rectangular backyard which is 2km long and 2km wide. They want to create a hill area so that they can have moto- cross fun in the backyard. More precisely, they want that there are four nicely spaced-out hills! You remember your MAT9004 lectures and the following functions comes to your mind (of course!): f(x, y) = x2y2e−x 2−y2 . Explain to your friend how the plot of this function could help them to model their backyard! Specifically, do the following: determine all stationary points of f(x, y) and classify those for which the Hessian test is not inconclusive; explain what this has to do with your friend’s design for the backyard. Note: For the classification of stationary points, you are not required to compute the Hessian matrix by hand, but you can simply use WolframAlpha to compute the required determinant values; you need to write down these values and explain what they tell you! Question 3.3 (5 marks, Lecture 15) You are doing some important data science stuff and need to compute the value of f(x, y) = 3xy + log3(y) at (a, b) = (3.002, 26.998). Unfortunately, you have misplaced your calculator and also WolframAlpha is offline! Luckily, you remember that MAT9004 taught you how to use first order approximations! Apply this and describe how you estimate the value of f(a, b). Having no calculator at hand, you must approximate ln(3) ≈ 1. Question 3.4 (10 marks, Lecture 18) You’re the CEO of Cake-X, a new cake factory! The secret of your success is that you can produce a large number of different cakes based on some simple doughs. Specifically, every cake is made by using a certain number of units of dough, where each unit is one of the following: • blue dough (5 different flavours), • orange dough (10 different flavours), • yellow dough (8 different flavours). The following gives a description of different production procedures. It will be your task to decide how many different cakes can be produced with each procedure. You have to use a certain counting formula and you have to explain why you think that the formula you have used is the correct one. Your final answer for each case should be an explicit integer. a) Procedure 1 “greeny”: a cake is prepared by mixing 4 different units of doughs, with at most one unit of orange dough. b) Procedure 2 “small tower”: a cake is prepared by having 3 layers (bottom, middle, and top layer) such that each layer contains one unit of dough and there is no repetition of colours. c) Procedure 3 “orange bonanza”: a cake has a bottom layer that is made by mixing 3 units of orange dough, and a top layer that is made by mixing 3 units of yellow dough of different flavours. Note: Your instructor is not a cake chef, so following these recipes is at your own risk. Question 3.5 (4 marks, Lecture 20) A chocolate factory produces pralines with and without peanuts. Since both are produced in the same facility, it happens that 0.1% of all “peanut-free pralines” are contaminated and contain traces of peanuts. The company developed a test that can determine whether a “peanut-free praline” is contaminated. The test is rather good because out of 1000 contaminated pralines, the test usually detects 950 of them. However, out of 100 pralines that are not contaminated, the test usually (falsely) reports that 10 of these are contaminated. Now here’s the question: Suppose the test says that a randomly selected praline is contaminated; what’s the probability that this praline is actually contaminated? Question 3.6 (7 marks, Lecture 21) Rody and Cody, two eager MAT9004 students, are playing a game on the following table of numbers: 4 2 -3 -3 -1 1 Rody chooses a row R and Cody chooses a column C, and they reveal their choices simultaneously. The number in the chosen row and column is the number of homework questions that Cody must help Rody with. A negative number −x means Rody must help Cody with x questions. (Clearly, Rody and Cody have not carefully read our policies because this would be a serious case of collusion, if not plagiarism!) Suppose that Rody chooses R randomly, choosing the first row with probability 2/3 and the second row with probability 1/3, and Cody chooses the column C at random, choosing the first and second with probability 1/4 each and the third with probability 1/2. Let X be the random variable that describes the value of the number in row R and column C. Let Y be the random variable with Y = −1 if X < 0 and Y = 1 if X > 0. a) Write down the probability distribution of X and determine E[X]. b) Determine Var[X]. c) Determine Pr[Y = 1]. d) Are Y and R independent? e) Are Y and C independent? Question 3.7 (5 marks, Lecture 21) Let a < b be unknown parameters and consider a function fa,b : R→ R with fa,b(x) = { x− 1 (x ∈ [a, b]) 0 (x /∈ [a, b]). Note: Here the subscripts a, b of fa,b are not partial derivatives, but just subscripts to indicate that the definition of this function depends on these two parameters. Your task is to find out for which a and b the function fa,b is a PDF of a continuous probability distribu- tion: determine a function g : D → R such that fa,b is a PDF if and only if a ∈ D and b = g(a). Question 3.8 (5 marks) This question has to be answered online in Moodle: please go to Online Quiz 3. Follow the instructions carefully; this question will be marked automatically and if you choose the wrong input format, the system will mark the question with 0 marks.
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