# SOCR EduMaterials Activities JointDistributions

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## 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 $$Y$$.

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