Introduction to Probability Theory

Computer Lab 7: Joint distribution functions

Consider the experiment of rolling two balanced dice, and define two random variables:

• X is the sum of the two numbers,
• Y is the difference of the two numbers.

1. Joint probability mass function
   
   • Use the computer to create a large random sample for the experiment of rolling the two dice (in MINITAB, select Calc -> Random Data -> Integer... and create at least 10000 rows of numbers in columns C1 and C2 to simulate at least 10000 outcomes). For each outcome, calculate the values of the random variables X and Y.

   • Use Stat -> Tables -> Cross Tabulation... (only display Total percents) to obtain estimates of the probabilities Pr(X = x ∩ Y = y) for x = 2, 3, ..., 12 and y = -5, -4, ..., 5.

   • Use this information to fill out the above table.

   • We define the joint probability mass function of X and Y by
     
     f (x,y) = Pr(X = x ∩ Y = y).

2. Marginal distributions
   • Fill out the column and row labeled "Total" in the above table.
   • We define the marginal probability mass function of X and Y as
     
     f₁(x) = Pr(X = x), and f₂(y) = Pr(Y = y).
     Use the table above to estimate the following values

     • f₁(6) = Pr(X = 6) =

     • f₂(3) = Pr(Y = 3) =

     Which part of the table has information on f₁?

     Which part of the table has information on f₂?

3. Independence
   • Are the events (X = 6) and (Y = 3) independent? Why or why not?

     
     

   • More generally, are the random variables X and Y independent? Why or why not?