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

1. Briefly explain why there are 27 possible outcomes.
2. Show that $\text{P}(X=1,Y=0)=2/27$ by listing the outcomes that comprise the event $\{X=1,Y=0\}$.
3. Show that $\text{P}(X=1,Y=3)=1/27$ by listing the outcomes that comprise the event $\{X=1,Y=3\}$.
4. Show that $\text{P}(X=2,Y=0)=6/27$ by listing the outcomes that comprise the event $\{X=2,Y=0\}$.
5. Make a table representing the marginal distribution of $X$ and compute $\text{E}(X)$.
6. Make a table representing the marginal distribution of $Y$ and compute $\text{E}(Y)$.
7. Find the conditional distribution of $Y$ given $X=x$ for each possible value of $x$.
8. Make a table representing the distribution of $\text{E}(Y|X)$.
9. Find the conditional distribution of $X$ given $Y=y$ for each possible value of $y$.
10. Make a table representing the distribution of $\text{E}(X|Y)$.
11. 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.
    1. Method 1: simulate outcomes from the probability space (i.e., prizes in the packages)
    2. Method 2: simulate an $(X,Y)$ pair directly from the joint distribution (without simulating outcomes from the probability space)
    3. Method 3: simulate an $(X,Y)$ pair by first simulating $X$ from directly from its marginal distribution (without simulating outcomes from the probability space).
12. 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.
    1. $\text{P}(X=3)$
    2. the marginal distribution of $X$
    3. $\text{E}(X)$
    4. $\text{Var}(X)$
    5. $\text{P}(X=2,Y=1)$
    6. $\text{E}(XY)$
    7. $\text{Cov}(X,Y)$
    8. $\text{P}(X=1|Y=0)$
    9. the conditional distribution of $X$ given $Y=0$
    10. $\text{E}(X|Y=0)$
    11. $\text{P}(Y=0|X=1)$
    12. the conditional distribution of $Y$ given $X=1$
    13. $\text{E}(Y|X=1)$