Final answer:
The joint PMF (Probability Mass Function) of X and Y can be found by considering all the possible outcomes of tossing a balanced die twice. We can create a table to show the probabilities of each outcome.
Step-by-step explanation:
The problem is related to finding the joint PMF (Probability Mass Function) of the variables X and Y when tossing a balanced die twice, where X and Y represent the smaller and larger results of the die tosses, respectively. By considering a six-sided die, there are a total of 36 outcomes when rolling the die twice. To construct the joint PMF, we enumerate all possible pairs (x, y), where 1 ≤ x ≤ y ≤ 6. For example, if the outcome is (2,4), X would be 2 and Y would be 4.
Let's denote the PMF as p(x,y). Because the die is fair, each outcome of the two dice has a probability of 1/36. For a valid (x, y) pair, the number of outcomes that satisfy these X and Y values depends on whether x equals y or not. If x = y, there's only one way to achieve this outcome (rolling the same number twice). However, if x ≠ y, there are two distinct outcomes that produce the pair (because the two rolls could happen in either order). Therefore, for x ≠ y, the probability is 2/36 or 1/18, and for x = y, the probability is 1/36.
To construct the PMF table, list all possible values that X and Y can take and their associated probabilities. The values that Y can take on are 1-6 inclusive, and the same for X, keeping in mind that X ≤ Y for all pairs. The PMF table would look like a triangular table with the probabilities decreasing down the diagonal where x = y.