Introduction to Probability Theory

Computer Lab 8: Conditional Distributions

In this lab, we will examine conditioning of binomial random variable.

1. Probability of success on the first trial conditioned on the total number of successes.
   • Use MINITAB to create 10000 rows of random numbers chosen from a binomial random variable with n = 1, and p, a probability of your choice between 0 and 1. Store the results in C1 and give this column the title X.
   • Repeat the above with 10000 rows of random numbers chosen from a binomial random variable with n =4, and p, the same choice as above. Store the results in C2 and give this column the title Y-X.
   • Add columns 1 and 2, store it in C3 and give this column the title Y.
   • What distribution does the random variable Y have? Check this by taking a tally of the Y random variable and compare this against the theoretical probabilities.
   • Use Stat->Table-> Cross Tabulation... with row variable X and column variable Y to tabulate the column percents.
   • Describe what you see.

     
     

     
     

     
     

     
     

     
     

     
     

     
     

     
     

   • Repeat the steps above changing the value of p and describe you results. Share you results with someone else.

     
     

     
     

     
     

     
     

     
     

     
     

     
     

     
     

   • Are the two tables similar?

     
     

     
     

     
     

     
     

   • Choose a possible value y for Y and compute Pr{Y=y|X=1}. Compare this theoretical answer with the answer obtained in the table.

     
     

     
     

     
     

     
     

2. Probability of success on the first few trials conditioned on the total number of successes.
   • Use MINITAB to create 10000 rows of random numbers chosen from a binomial random variable with n = 3, and p, a probability of your choice between 0 and 1. Store the results in C1 and give this column the title X.
   • Repeat the above with 10000 rows of random numbers chosen from a binomial random variable with n, a number of your own choosing, and p, the same choice as above. Store the results in C2 and give this column the title Y-X.
   • Add columns 1 and 2, store it in C3 and give this column the title Y.
   • Use Stat->Table-> Cross Tabulation... with row variable X and column variable Y to tabulate the column percents.
   • Describe what you see.

     
     

     
     

     
     

     
     

     
     

     
     

     
     

     
     

   • Repeat the steps above changing on the value of p and describe you results. Share you results with someone else.

     
     

     
     

     
     

     
     

     
     

     
     

     
     

     
     

   • Are the two tables similar?

     
     

     
     

     
     

     
     

   • Choose two possible values x for X and y for Y and compute Pr{Y=y|X=x}. Compare this theoretical answer with the answer obtained in the table.