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Chapter 3: Problem 11

Using the computer, obtain plots of the pdfs of chi-squared distributions with degrees of freedom \(r=1,2,5,10,20\). Comment on the plots.

Short Answer

Expert verified
Upon generating and plotting the chi-squared distributions for the specified degrees of freedom (DoF), it was observed that as the DoF increased, the distribution changed from being highly skewed to the right (DoF 1) to becoming approximately symmetrical (DoF 20). This suggests that as the degrees of freedom increase, the chi-squared distribution approximates a normal distribution more closely.

Step by step solution

01

Generation of Chi-Squared Distributions

For each specified degree of freedom (1, 2, 5, 10, 20), use a statistical function to generate a Chi-Squared distribution. In Python, you can use the 'numpy.random.chisquare' method or the 'scipy.stats.chi2' function in the scipy library.
02

Creation of Probability Density Functions (PDFs)

Create the PDFs for each of the generated Chi-Squared Distributions. The scipy library in Python has a method 'pdf' for creating PDFs. You will pass in the random variables from the chi-squared distribution and the degrees of freedom to this function.
03

Plotting the PDFs

Generate plots for each of the PDFs created in Step 2 to visually represent the data. Use a plotting library such as matplotlib in Python. You might consider using subplots to easily compare the PDFs.
04

Analysis and Interpretation

Analyze each PDF plot individually, and then compare amongst each other. You'll observe that as the degree of freedom increases, the shape of the distribution changes. It moves from left skewness to becoming approximately symmetrical.

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