Introduction to Probability Theory

Computer Lab 8: Random numbers

In this lab, we will see how to plot an approximation of the p.d.f. of a continuous random variable, given a large random sample for this random variable. We will also see how to generate random numbers according to a given continuous distribution.

1. How to plot an approximation of the p.d.f. of a continuous random variable
   • Use MINITAB to create 10000 rows of random numbers from a normal distribution with mean 0 and variance 1. Store the data in Column C1.
   • Repeat the above with 1000 rows of data, and store the results in Column C2.
   • Use Graph -> Histogram... to plot a histogram of the data in Columns C1 and C2. Click on Frame -> Multiple Graphs... and choose the option Overlay graphs on the same page, to see both graphs at the same time.
   • Are the two plots similar? Why or why not? Can you see why there is a problem?

     
     

     
     

     
     

     
     

   • Plot a histogram again, but now click on Options... and select Percent under Type of Histogram. Is the result satisfactory? Why or why not?

     
     

     
     

     
     

   • Repeat the above with Density selected under Type of Histogram. Is the result satisfactory? Why or why not?

     
     

     
     

     
     

   • Based on the above results, what do you think is the best way to plot an approximation of the p.d.f. of a random variable, given a large random sample for this random variable?

     
     

     
     

     
     

     
     

2. How to generate random numbers according to a given continuous distribution
   Suppose we want to create random numbers according to a triangular density function, supported on the interval [-1,1].
   • Draw a picture showing the graph of the p.d.f. of the corresponding random variable (call it X), and find a formula for this p.d.f.

     
     

     
     

     
     

     
     

     
     

     
     

   • Find a formula for the c.d.f. F of X and plot this function.

     
     

     
     

     
     

     
     

     
     

     
     

   • If Y = F ( X ), find X in terms of Y.

     
     

     
     

     
     

     
     

     
     

     
     

   • We will see that Y has a uniform distribution on [0,1]. Use this information to generate a random sample for X of size 10000.

     
     

     
     

   • Check your result by using this random sample to plot an approximation of the p.d.f. of X.

     
     

     
     

3. Reading Assignment: Section 3.8