Assist me to solve the following questions (1-4) in plain sheet papers

W Home P 2023T2SIT787_Assignment3_newv2.pd .P SIT787 T1 2024 Assessment 3 ( @ X 21 30 Upgrade now X = Menu Home Edit Comment Convert Page Fill & Sign Protect Tools Batch & WPS AI Share ... 19 A- IAI Q O Edit PDF . Add Text Add Picture PDF to Word PDF to Image Highlight Text comment Text Box Sign OCR PDF Extract Text Snip and Pin Find and Replace Auto Scroll Eye Protection Mode Sync Translate W SIT787 Mathematics for Al Assignment 3 2023 Tri 1 Page 2 of 7 Question 1) Graph Theory: Consider the following incidence matrix of a graph G = (V, E) with V = {a, b, c, d'} and {e1, ez, e3, e4, es, ed} DY e1 ez e3 e4 es e6 BE a 1 -1 0 -1 0 -1 1 -1 0 0 0 E M = 1 10 d 1 -1 0 -1 Based on the information you obtain from the incidence matrix M, answer these questions: (? (a) What type of graph does M represent? b) Find the adjacenty matrix A for this graph. (c) Draw the graph. (d) How many paths of length 2 are there between nodes b and c (without direct counting)? (e) In terms of connectivity of the graph, what is your interpretation of tr(AZ)? [The question is not about the value.] (f) Without direct calculations, find one of the eigenvalues of A based on the information you can get from A. Then calculate its corresponding eigenvector. [2+2+2+2+2+5=15 marks] Navigation K > Back to Page 1 1 83.F Mostly sunny Q Search O X 9:07 AM 5/27/2024W Home P 2023T2SIT787_Assignment3_newv2.pd .P SIT787 T1 2024 Assessment 3 ( @ X [21 30 Upgrade now X = Menu Home Edit Comment Convert Page Fill & Sign Protect Tools Batch & WPS AI Share ... 19 A- IAI Q O Edit PDF . Add Text Add Picture PDF to Word PDF to Image Highlight Text comment Text Box Sign OCR PDF Extract Text Snip and Pin Find and Replace Auto Scroll Eye Protection Mode Sync Translate W LAW SIT787 Mathematics for Al Assignment 3 2023 Tri 1 Page 3 of 7 Question 2) Probability, Bayes' Theorem: A data scientist is interested in studying the relationship between having a social media account and obesity. He collected some data represented in the following table. 3E Individual |Having a social media account |Obesity F Individual1 Yes Yes Individual2 Yes Yes Individual3 Yes Yes Individual4 No No Individual5 No Yes (? (a) Find the probability distributions of having a social media account and obesity. (b) Find the joint probability distribution of having a social media account and obesity. (c) Are these two distributions independent? (d) The data scientist is interested in whether having a social media account helps gain in- formation about obesity. The concept of mutual information quantifies how one feature is related to another feature. The mutual information is defined as follows for two random variables X and Y . EE mutual information = Px, Y ( x, y ) In P xY ( x, y ) Px(x)py (y) Suppose the joint probability distribution of having a social media account and obesity is shown by Px,y (x, y) and the marginal distributions of having a social media account and obesity are px(x) and py (y) respectively. Compute the mutual information between having a social media account and obesity. [You can use a calculator to find In(x).] (e) Theoretically, what is the mutual information of two independent random variables? [5+5+5+5+5= 25 marks] Navigation K > Back to Page 3 1 83.F O 9:09 AM Mostly sunny Q Search 109+ 5/27/2024W Home B3 2023712511787 Assignment3_newvzpc [ SIT787 T1 2024 Assessment 31 ) %X + 2 @ = Menu 5 a9 v Home Edit Comment Convert Page Fill & Sign Protect Tools Batch & A A2 1T 82 & B B @Em 2 8B =m @ Q @ & EditPDF~ Add Text Add Picture PDF toWord~ PDF to Image Highlight~ Text comment Text Box Signv OCRPDF ExtractTextv Snip and Pin Find and Replace Auto Scroll Eye Protection Mode ~ H O Question 3) Probability, Distributions: Let X be a discrete random variable that takes values in {2,1,0,1,2} with equal probability. Also, Y is another discrete random variable M) : defined as Y = |X]|. [0 (a) Construct the joint probability distribution table. Are X and Y independent? g Justify. (b) Find Corr(X, Y). (c) Based on your answer to part (a), can you explain the result in part (b)? (d) Although you have solved similar questions in questions 2 and 3, can you explain the fundamental difference between these two questions? [5+5+5+5= 20 marks] [(J Navigation K 3/7 > >l BacktoPage3? O s2F % Mostly sunny i= Q Search & L D '? @ ; @ E'S f C' @g 8 Sync Translate =) o o 910AM 5/21/2024 ra o & & W Home P 2023T2SIT787_Assignment3_newv2.pd .P SIT787 T1 2024 Assessment 3 ( @ X 21 30 Upgrade now X = Menu GASSOCY Home Edit Comment Convert Page Fill & Sign Protect Tools Batch & WPS AI Share ... 19 A- IAI Q O Edit PDF . Add Text Add Picture PDF to Word PDF to Image Highlight Text comment Text Box Sign OCR PDF Extract Text Snip and Pin Find and Replace Auto Scroll Eye Protection Mode Sync Translate mulupiers W Question 4) Now, let's solve some optimisation problems. (a) Linearise the function f ( x , y) = 2x + 3y - PY at the point (3, 1). 3E (b) Find the second order Taylor polynomial for f(x, y) = esx In(1 + y) at the point (0, 0). (?) (c) For the multivariate function f ( x, y, z) = yx2 + zy2 + z2 - 2yx + 2zy + y- z (i) Find all the stationary points(s) of this function. (ii) Find the Hessian matrix. (ii) Classify the stationary point(s). (d) Find all values for k so that f(x, y) = x# + kxy + y# has a local minimum at (1, 1). Give your answer in the form of an interval. (e) Find the maximum and minimum values of f (x, y, z) = -x - y + z subject to x2 + y" = 2 and x + y + z = 1. [ 5+5+[5+5+5]+5+10 = 40 marks] Navigation K