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.

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

Question 2:

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

and ?2=9

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

Question 3:

- 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.
- 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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