8. A tetrahedron shaped die gives one of the numbers 1, 2, 3, 4 with equal probabilities. We roll
two of these dice and denote the two outcomes by X1 and X2. Let Y1 =(minpX1, X2) and
Y2 = |X1 - X2|.
(a) Find the joint probability mass function of X1 and X2.
(b) Find the joint probability mass function of Y1 and Y2.
(c) Compute E[Y1Y2]
To solve this problem, we need to go step by step. Let's start with part (a) and find the joint probability mass function (PMF) of X1 and X2.
(a) Joint Probability Mass Function (PMF) of X1 and X2:
Since each die has four equally likely outcomes (1, 2, 3, or 4), the probability of each outcome is 1/4.
To find the joint PMF, we need to consider all possible outcomes of rolling two dice and their corresponding probabilities.
Let's create a table to list all possible outcomes of (X1, X2) and their probabilities:
| X1 | X2 | Probability |
|--------|--------|-------------|
| 1 | 1 | 1/4 * 1/4 |
| 1 | 2 | 1/4 * 1/4 |
| 1 | 3 | 1/4 * 1/4 |
| 1 | 4 | 1/4 * 1/4 |
| 2 | 1 | 1/4 * 1/4 |
| 2 | 2 | 1/4 * 1/4 |
| 2 | 3 | 1/4 * 1/4 |
| 2 | 4 | 1/4 * 1/4 |
| 3 | 1 | 1/4 * 1/4 |
| 3 | 2 | 1/4 * 1/4 |
| 3 | 3 | 1/4 * 1/4 |
| 3 | 4 | 1/4 * 1/4 |
| 4 | 1 | 1/4 * 1/4 |
| 4 | 2 | 1/4 * 1/4 |
| 4 | 3 | 1/4 * 1/4 |
| 4 | 4 | 1/4 * 1/4 |
So, the joint PMF of X1 and X2 is:
P(X1, X2) = [1/4 * 1/4 for all possible outcomes]
(b) Joint Probability Mass Function (PMF) of Y1 and Y2:
To find the joint PMF of Y1 and Y2, we need to determine the possible outcomes and their corresponding probabilities.
Y1 = min(X1, X2) and Y2 = |X1 - X2|
Based on these definitions, here are all possible outcomes of (Y1, Y2) and their probabilities:
| Y1 | Y2 | Probability |
|--------|--------|-------------|
| 1 | 0 | P(X1=1, X2=1) |
| 1 | 1 | P(X1=1, X2=2) + P(X1=2, X2=1) |
| 1 | 2 | P(X1=1, X2=3) + P(X1=3, X2=1) |
| 1 | 3 | P(X1=1, X2=4) + P(X1=4, X2=1) |
| 2 | 0 | P(X1=2, X2=2) |
| 2 | 1 | P(X1=2, X2=3) + P(X1=3, X2=2) |
| 2 | 2 | P(X1=2, X2=4) + P(X1=4, X2=2) |
| 3 | 0 | P(X1=3, X2=3) |
| 3 | 1 | P(X1=3, X2=4) + P(X1=4, X2=3) |
| 4 | 0 | P(X1=4, X2=4) |
To calculate the probabilities, substitute the corresponding values from the joint PMF of X1 and X2.
(c) Compute E[Y1Y2]
To compute the expected value E[Y1Y2], we need to find the product of Y1 and Y2 for each possible outcome and calculate the weighted sum of these products using the joint PMF of Y1 and Y2.
E[Y1Y2] = Σ(Y1Y2 * P(Y1, Y2))
Substitute the values of Y1 and Y2 from the joint PMF of Y1 and Y2 and their corresponding probabilities. Then, sum up the products to get the expected value.