# Random Variables: Joint, Conditional, and Marginal Distributions

The latest series of collectible Lego Minifigures contains 3 different Minifigure prizes (labeled 1, 2, 3). Each package contains a single unknown prize. Suppose we only buy 3 packages and we consider as our sample space outcome the results of just these 3 packages (prize in package 1, prize in package 2, prize in package 3). For example, 323 (or (3, 2, 3)) represents prize 3 in the first package, prize 2 in the second package, prize 3 in the third package. Let \(X\) be the number of distinct prizes obtained in these 3 packages. Let \(Y\) be the number of these 3 packages that contain prize 1. Suppose that each package is equally likely to contain any of the 3 prizes, regardless of the contents of other packages; let \(\text{P}\) denote the corresponding probability measure.

It can be shown that the joint distribution of \(X\) and \(Y\) can be represented by the following table.

\(\text{P}(X = x, Y=y)\) | ||||
---|---|---|---|---|

\(y\) | ||||

\(x\) | 0 | 1 | 2 | 3 |

1 | 2/27 | 0 | 0 | 1/27 |

2 | 6/27 | 6/27 | 6/27 | 0 |

3 | 0 | 6/27 | 0 | 0 |

- Briefly explain why there are 27 possible outcomes.
- Show that \(\text{P}(X = 1, Y = 0) = 2/27\) by listing the outcomes that comprise the event \(\{X = 1, Y = 0\}\).
- Show that \(\text{P}(X = 1, Y = 3) = 1/27\) by listing the outcomes that comprise the event \(\{X = 1, Y = 3\}\).
- Show that \(\text{P}(X = 2, Y = 0) = 6/27\) by listing the outcomes that comprise the event \(\{X = 2, Y = 0\}\).
- Make a table representing the marginal distribution of \(X\) and compute \(\text{E}(X)\).
- Make a table representing the marginal distribution of \(Y\) and compute \(\text{E}(Y)\).
- Find the conditional distribution of \(Y\) given \(X=x\) for each possible value of \(x\).
- Make a table representing the distribution of \(\text{E}(Y|X)\).
- Find the conditional distribution of \(X\) given \(Y=y\) for each possible value of \(y\).
- Make a table representing the distribution of \(\text{E}(X|Y)\).
- Describe three methods for how you could use physical objects (e.g., cards, dice, spinners) to simulate an \((X, Y)\) pair with the joint distribution given by the table above.
- Method 1: simulate outcomes from the probability space (i.e., prizes in the packages)
- Method 2: simulate an \((X, Y)\) pair directly from the joint distribution (without simulating outcomes from the probability space)
- Method 3: simulate an \((X, Y)\) pair by first simulating \(X\) from directly from its marginal distribution (without simulating outcomes from the probability space).

- Suppose you have simulated many \((X, Y)\) pairs. Explain how you could use the simulation results to approximate each of the following. You should not do any of the calculations; rather, explain in words how you would use the simulation results and simple operations like counting and averaging.
- \(\text{P}(X = 3)\)
- the marginal distribution of \(X\)
- \(\text{E}(X)\)
- \(\text{Var}(X)\)
- \(\text{P}(X = 2, Y = 1)\)
- \(\text{E}(XY)\)
- \(\text{Cov}(X, Y)\)
- \(\text{P}(X = 1 | Y = 0)\)
- the conditional distribution of \(X\) given \(Y=0\)
- \(\text{E}(X|Y=0)\)
- \(\text{P}(Y = 0 | X = 1)\)
- the conditional distribution of \(Y\) given \(X=1\)
- \(\text{E}(Y|X=1)\)