Write a python code find the maximum of a function. One of them is visual using graphics, another one is analytical by differentiating the function.

INSTRUCTIONS TO CANDIDATES

ANSWER ALL QUESTIONS

Question 1:

Download the file from the following web location (URL:) http://hyperion.usc.edu/pressure.txt Load the file into your working environment (a jupyter notebook) and answer the following questions with reference to this file.

1. How many columns of data does the file have, 
2. How many rows of data does it have ?
3. Evaluate the mean and variance for each column.
4. plot the first column again the column before last, 
1. using as label for the x-axis: "Time (sec)" 
2. and label for the y-axis: "pressure (MPa)", 
3. and as title: "Pressure at First Sensor".

Question 2:

1. Sample 100 points from a normal distribution with parameters ?=1

 and ?2=9

1. , using the scipy.stats package.
2. Plot the pdf of these samples.
3. Fit a normal distribution to these samples and print the parameters of the fit.
4. Fit a beta distribution to these samples and print the parameters of the fit.
5. For the beta distribution in question 4., evaluate the mean and variance using the mean() and var() methods in scipy.stats.
6. Repeat questions 1-6 above using 10,000 samples (instead of 100 samples).
7. Comment on the results of your calculations.

Question 3:

1. Retrieve the 30 samples of the modulus of elasticity that we obtained by taking secants of the three stress-strain curves, and plot the probability density function of these 30 samples using the gaussianMixture function we have been using in class; use two bandwidth or scale parameters equal to 0.5 and 1.5, on the same plot. There will be marks for clarity of the plots: labels on x and y axes, title and legend.
2. Using a beta distribution to describe this modulus, evaluate the maximum likelikhood estimate of the parameters of this distribution. Write a python code to do that. There are many ways to find the maximum of a function. One of them is visual using graphics, another one is analytical by differentiating the function, and a third is numerical using an optimization package from python). Any one of these methods is acceptable as part of your answer.

 

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