Submission Instructions
Please submit a report (PDF or Word) along with your MATLAB code to the Avenue.
Please compress the files into a zip file or other compression format supported by 7-zip. The file name should follow: lastname_studentnumber (ex. Milewski_00123456.zip)
Mark sure that your code is well commented. Suppress output as necessary to make the results as clear and easy as possible/practical for the TA to understand.
Question 1 [25 Marks]
• Create a program that draws numbers from a standard normal distribution by randomly generating a number from the uniform distribution U[0,1). Use can use the inverse standard normal function ‘norminv()’. Draw 100, 1000 and 10000 numbers using your program and plot the distribution against the parametric distribution. How well do the distributions fit the parametric distribution? [5 Marks]
• Using the generated numbers from part A, use the relationship to transform the generated numbers from N(0,1) to N(5,2). [5 Marks]
• Using the transformed generated numbers from part B, compute the 95th percentile range by counting 2.5% observations from the bottom and 2.5% observations from the top (computer is doing the counting). How well does this fit the parametric 95% percentile range? What are the sources of error? [15 Marks]
Question 2 [50 marks total]
• Create a program that uses Taylor series approximation of between -3 and 3 with and . Plot the errors, using , for first, second and third order approximations. What difference do you notice between the errors when h changes? When the order changes? What is the average error and how can this be misleading? [25 Marks]
• Change the program so that is updated with each new estimate for (recursively estimated) with . Plot the errors for the first and second order approximations. How do they compare the Part A? What are the sources of error? [25 Marks]
Question 3 [50 marks TOTAL]
Consider the function . Assume an error tolerance of .
• Find the equilibrium point where . Plot each iteration step and the 45-degree line. What do you notice? [25 Marks]
• Find the root of the equation, where , using the bisection, fixed-point iteration, and Newton-Raphson methods. For each method, plot the function and each iteration point on a graph. Describe the performance of each method. [25 Marks]