Introduction to Probability Theory

Computer Lab 9: The Normal Distribution
and the Central Limit Theorem

In this lab, we will get familiar with the normal distribution and use MINITAB to explore the Central Limit Theorem.

1. The normal distribution
   • Use MINITAB to plot the density function of a normal distribution with:
     • Mean 0 and standard deviation 1
     • Mean 1 and standard deviation 1
     • Mean 0 and standard deviation 0.3

   • Then answer the following questions:
     • What happens to the density function if you change the mean of the random variable but keep the standard deviation constant?

       
       

       
       

       
       

     • What happens to the density function when you change the standard deviation but keep the mean constant?

       
       

       
       

       
       

     • The density function of a normal random variable is bell-shaped.
       • For what value of x does the density function reach its maximum?

         
         

         
         

       • Where are the two inflection points of the density function?

         
         

         
         

2. The Central Limit Theorem
   The Central Limit Theorem says that the sample mean of a random sample of size n taken from a distribution of mean m and standard deviation s has a distribution which is approximately normal (i.e. which tends towards a normal distribution as n goes to infinity) with mean m and variance s²/n. We will use MINITAB to explore this statement.
   • Assume that X₁, X₂, ..., X_n are normally distributed with mean m and standard deviation s. Then, the sample mean (X₁ + X₂ + ... + X_n)/n is also normally distributed with mean m and variance s²/n. In the case of normally distributed random variables, this is true for any integer n. With MINITAB, create 10000 random samples of size 3 from a normal distribution with mean 0 and variance 3.
     Use Calc->Row Statistics->Mean... Calculate the sample mean in each case and plot a histogram approximating the distribution of the sample mean. Check that the approximate density function has a mean of 0 and a variance of 1.

     
     

     
     

     
     

   • Create 16 columns for random samples of size 100000 taken from a uniform distribution on the interval [0,1].
   • What is the mean and standard deviation of a random variable, uniform on [0,1]?
   • Give the column average of the first 4 columns and of all 16 columns.
   • Is the approximate density function for these column averages almost normal? Why or why not? Look at a histogram.

     
     

     
     

     
     

   • Give the mean and standard deviation of the column averages above.
   • Simluate 10000 binomial random variables having variance approximately 3. Standardize the random variable and store the values. Note that if X is binomial with parameters n and p, then E(X) = n p and Var(X) = n p (1 - p).

     
     

     
     

     
     

   • Repeat all of the above with np ≥ 10. What do you conclude? How is this an illustration of the Central Limit Theorem?