Introduction to Probability Theory

Computer Lab 7: Random numbers

In this lab, we will see how to plot an approximation of the density function 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 distribution 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 density function of a random variable, given a large random sample for this random variable?

     
     

     
     

     
     

     
     

2. Two views of the exponential random variable.
   • Choose a positive constant b.
   • Graph the distribution function F of an exponential random variable with parameter b is 0 if x is negative and 1 - exp(bx) if x is positive.

     
     

     
     

     
     

     
     

   • Find the inverse function for F.

     
     

     
     

     
     

     
     

   • Use MINITAB to create 10000 rows of random numbers from a exponential distribution with mean 1/b. Store the data in Column C3.
   • Use MINITAB to create 10000 rows of random numbers from a uniform distribution on [0,1]. Store the data in Column C4.
   • Let C5 = -LOGE(1-C4)/b
   • Overlay the histograms of C3 and C5 and compare.
   • Let Y be a uniform random variable on [0,1] and compute Pr{-ln(1-Y)/b < x}

     
     

     
     

     
     

     
     

     
     

     
     

     
     

     
     

3. How to generate random numbers according to a given continuous distribution
   Suppose we want to create random numbers according to a Pareto distribution function F(x) = 1 - (x/x₀)^a+1 supported on the interval [x₀,¥). Here both x₀ and a are positive. Let X be a random variable having this distribution.

   
   

   • Choose values for x₀ and a. Graph the distribution function F of X.

     
     

     
     

     
     

     
     

     
     

   • Find a formula for this density function and give a graph.

     
     

     
     

     
     

     
     

     
     

     
     

   • If Y = F ( X ), find X in terms of Y. In other words, find the inverse function for F.

     
     

     
     

     
     

     
     

     
     

     
     

   • 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 density function of X.