Question

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

The graphs show the distributions of chi-square values with different degrees of freedom. As the degrees of freedom increase, the peak of the distribution shifts to the right and pdf shape begins to look like a normal distribution.

Step-by-step solution

Import Required Libraries

First, import the necessary libraries. For instance, if using Python, you'll typically require 'matplotlib' for plotting the graphs and 'scipy.stats' for generating the chi-squared distributions.

Define Chi-Squared Distribution

Define the chi-squared distribution with varying degrees of freedom. In scipy library, you can use the 'scipy.stats.chi2' function to declare the chi-squared distribution.

Generate PDF and Plot

Now, generate the pdf for each degree of freedom and plot them. The pdf can be calculated using the 'pdf()' function, and then use 'matplotlib.pyplot.plot()' function to create the desired graph.

Observation

Observe and comment on the plots according to how the chi-squared distributions change with respect to the different degrees of freedom. As degrees of freedom increase, the peak of the pdf moves to the right and the distribution starts to resemble a normal distribution as per the central limit theorem.