# SOCR EduMaterials Activities JointDistributions

## This is an activity to explore the joint distributions of X and Y through two simple examples.

• Description: You can access the applets for the following experiments at SOCR Experiments
• Exercise 1: Die coin experiment:

A die is rolled and the number observed X is recorded. Then a coin is tossed number of times equal to the value of X. For example if X = 2 then the coin is tossed twice, etc. Let Y be the number of heads observed. Note: Assume that the die and the coin are fair.

• 1. Construct the joint probability distribution of X and Y.
• 2. Find the conditional expected value of Y given X = 5.
• 3. Find the conditional variance of Y given X = 5.
• 4. Find the expected value of Y.
• 5. Find the standard deviation of Y.
• 6. Graph the probability distribution of Y.
• 7. Use SOCR to graph and print the empirical distribution of Y when the experiment is performed
• a. n = 1000 times.
• b. n = 10000 times.
• 8. Compare the theoretical mean and standard deviation of Y (parts (4) and (5)) with the empirical mean and standard deviation found in part (8).

Below you can see a snapshot of the theoretical distribution of Y.

• Exercise 2: Coin Die experiment:

A coin is tossed and if heads is observed then a red die is rolled. If tails is observed then a green die is rolled. You can choose the distribution of each die as well as the probability of heads. Choose for the red die the 3-4 flat distribution and for the green die the skewed right distribution. Finally using the scroll button choose $p=0.2$ as the probability of heads. Let X be the score of the coin (1 for heads, 0 for tails), and let Y be the score of the die (1,2,3,4,5,6).

• 1. Construct the joint probability distribution of X,Y.
• 2. Find the marginal probability distribution of Y and verify that it is the same with the one given in the applet.
• 3. Compute E(Y).
• 4. Compute E(Y) using expectation by conditioning E[E(Y | X)].
• 5. Run the experiment 1000 times take a snapshot and comment on the results.

Below you can see a snapshot of the theoretical distribution of Y.