1. In this lab session, we will complete the introduction to SHAZAM by going over the remainder of the instructions on the handout entitled "How to Run Basic Types of Shazam Programs..." that was distributed in the lecture. In particular, our first session did not cover the procedure for using SHAZAM in conjunction with GNUPLOT to get laser printouts of nice plots. We will address this task in this lab session.
   
   
2. We will execute a "feet first" plunge into an example of using SHAZAM to plot the shapes of common distributions with different parameter values. We will look briefly at the tasks performed in the program entitled n:plotdist.sha. Do not feel uncomfortable about the complexity of the program. You will understand it by then end of the quarter. Now, there will be only a cursory overview of what is IN it, and we will concentrate on passive consumption of what it produces for us.
   
   
3. You may try any on-line self-grading quizzes you have not yet attempted.

If all you wanted to do was plot distributions, there are better graphics programs you might want to use. Shazam can be programmed to plot distributions however, so I have done this. The exercise will introduce you to the concept of "subroutines" or "macros" or "procedures." These are blocks of commands that take as input a number of parameters, can then be executed for those particular parameter values, and produce some sort of output. You need not issue the entire block of commands each time you want to do something similar. Instead, you set the values of the input parameters as desired, then just "execute" the procedure.

1. Use TED, or even Notepad, to look at the program file and review the plotdist.sha commands.
   
   
2. Note that there are some initial default settings for the input parameters, so that you can run the procedures under these settings and see "initial" plot shapes. The defaults are N(0,1), t(30), F(4,40), and chi-squared(4). Also initially, something called "scale" is set to 1 for "on," and maxf, maxchi, minn, and maxn have default settings.
   
   
3. Enter the SHAZAM program and type the command file 5 n:plotdist.sha if you are working in the lab. If you are working from home, and you have your own copy of the plotdist program, you will need to change the drive and directory appropriately.
   
   
4. To look at the default normal distribution: exec ndistn. Then plot a nice picture of this normal probability density function: plot normpdf x / gnu line. Then look at the corresponding cumulative density function: plot normcdf x / gnu line. You might then want to look at both of these functions on the same plot: plot normcdf normpdf x / gnu line.
   
   
5. Now consider a different normal distribution. Change the mean and standard deviation parameters using the gen1 command: gen1 mu=10, then gen1 sig=3.5. Now create the data for plotting this distribution: exec ndistn. Then plot the results: plot normcdf normpdf x / gnu line. What do you notice about the shapes of these curves compared to the previous ones? What do you notice about the spread? About the height of the curves at the mean?
   
   
6. Now turn off the automatic scaling of the axes: gen1 scale=0, and rerun the procedure: exec ndistn. Now plot the probability density function: plot normpdf x / gnu line. This is the default range, which can be changed to anything else using commands such as: gen1 minn=0, and gen1 maxn=50. Try this, then run the procedure again with exec ndistn. Now look at the pdf: plot normpdf x / gnu line.
   
   
7. Now clean up what you have done with the normal distribution by putting back its parameters and rerunning it: gen1 scale=1, gen1 mu=0, gen1 sig=1. Then exec ndistn. This restores the standard normal. Now save the standard normal probability density function with genr nor01=normpdf.
   
   
8. Try a t-distribution with 30 degrees of freedom (the default): exec tdistn. Plot the results: plot tpdf t / gnu line. Now save this curve by renaming to something that won't be overwritten next time you execute the procedure: genr t30=tpdf.
   
   
9. Try a t-distribution with fewer degrees of freedom, say 5: gen1 df=5. Rerun the procedure: exec tdistn. Plot the results: plot tpdf t / gnu line. Rename to save for later: genr t5=tpdf.
   
   
10. Now compare the t-distributions with 30 and 5 degrees of freedom on the same plot: plot t30 t5 t / gnu line.
    
    
11. Now try a t(100) distribution and compare to a standard normal: gen1 df=100. Then exec tdistn. Then genr t100=tpdf. Finally, plot t100 nor01 t / gnu line. What do you notice about these two distributions? If you like, try the same comparison with the t30 or t5 distributions that remain in the program's memory at this point. What do you notice?
    
    
12. Try the default F-distribution, with "4 and 40" degrees of freedom. Use exec fdistn. Then plot fpdf f / gnu line. What is different about the shape of this distribution, compared to the normal and the t?
    
    
13. Now change the value of maxf: gen1 maxf=2. Then exec fdistn. Then redo the plot: plot fpdf f / gnu line. What has happened?
    
    
14. Lastly, take a look at the shape of a Chi-squared distribution with four degrees of freedom (the default). Use exec chidistn. Then plot chipdf chi / gnu line. What do you observe about the shape of this chi-squared distribution? Save this curve by renaming: genr chi4=chipdf
    
    
15. Now change the degrees of freedom to 2 with gen1 k=2. Rerun the procedure with exec chidistn, and do the plot: plot chipdf chi / gnu line. What has changed? Save this curve by renaming: genr chi2=chipdf. Compare the two with: plot chi4 chi2 chi / gnu line. What difference does it make to the distributions to change the degrees of freedom?
    
    

At your leisure (what's that?) you are welcome to explore the shapes of these useful distributions for other parameter values. In the meantime, I am planning for a Java Applet that will show these ideas with less effort on the student's part. It might take a while, however.