Assignment2
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This document outlines an assignment to analyze two confidential datasets and simulate additional data with similar characteristics. Students are told the first dataset follows a Poisson process but their analysis shows otherwise. The second dataset is related to the first in some binomial fashion. Students must determine the true distributions of the datasets, show their relationship, and use this to simulate new matching data. Their friend provides some helpful hints, like considering waiting times and plotting the relationship between the two datasets.
Assignment2
- 1. KME272 Engineering Mathematics 2B 2019 : Assignment 2 Due 5pm September 4th; worth 7.5%. Submit online only. You have been asked to look at these highly confidential data and have been told certain things about it: 1. Data 1 correspond to a Poisson Process 2. Data 2 correspond to a Binomial Process based on Data 1 values Your job for $15,000 is to: 1. Find how the Poisson Process gives you the data you saw 2. Find the relationship between Data1 and Data2 3. Simulate data that behave like the data provided This awful task is helped by your busy statistics friend, who having briefly glanced at the data, makes the following obser- vations: 1. It’s clearly not a Poisson process, but maybe there’s a Poisson process involved. 2. Start by showing it’s not Poisson. 3. Maybe it’s to do with waiting times. 4. Try fitting some distributions so they have the same mean and variance as those data. 5. You’ll need to use the inverse transform method to simulate from the data. 6. Data2 looks like its values are high when the Data1 values are high - what’s going on with that? Can you plot this relationship? 1
- 2. 2 Tasks 1. Why is it “clearly not” Poisson? (a) Calculate summary statistics for Data1 and use them to argue that the distribution is not a Poisson distribution. 2 (b) Use the method of moments to estimate what the parameter of a Poisson distribution would be to give you those values. 2 (c) Collate how many values there are in the range 0-4, 5-9, 10-14, etc. and plot the resulting histogram. 2 (d) Use the Chi-square test to determine whether the data fit a Poisson distribution. 2 2. Find a better distribution for Data1: (a) Show your reasoning for which distribution you choose, and remember, you may need to compare several distributions. 4 (b) Give all measures of fit you use, including at least the Chi-square measure of fit, calculated as you did in the question above. You are encouraged to use other measures also. 4 3. You have been told Data2 has some kind of binomial relationship with Data1 values: (a) Why is it binomial? Provide a justification of why these data fit a binomial distribution. 2 (b) Provide plot(s) to show that Data2 follows a binomial distribution as a function of the values of Data1. 2 4. Simulate data using a spreadsheet or R that have the same characteristics as given in Data2 (which will require you work out the relationship between Data2 and Data1). In particular, the simulated data should 5 (a) have approximately the same mean and variance as Data2; (b) have the same upper and lower limits as Data2. 5. Comment on the processes that might be going on. Think about what the assumptions are of the distribution you have found that you think fits the Data1 and Data2 the best. 5 Style Guide 1. Submit your report as a pdf, and attach all other working (e.g., spreadsheet, R scripts). 2. Write your report in clear English, with correct spelling and grammar. 3. Lay your report out logically and clearly so information is easy to find. 4. Give full working and provide clear, fully labelled plots for your answers in your report. 5. Use full sentences for all answers: instead of writing “mean: 5” write “The mean value was 5” or similar. Total marks: 30