Question

Consider four independent rolls of a 6-sided die. Let X be the number of l's and let y be the number of 2's obtained. What is the joint PMF of X and Y?

Answer 1

The joint PMF of X and Y can be determined by considering all possible outcomes when rolling the die four times. By calculating the individual probabilities of X and Y and multiplying them together, we can determine the joint PMF for each combination of X and Y.

Step-by-step explanation:

The joint PMF (Probability Mass Function) of X and Y can be determined by considering all possible outcomes when rolling the die four times. Let's break down the calculation step by step:

1. X represents the number of l's obtained, and Y represents the number of 2's obtained. Since the die has six sides, each with an equal probability of 1/6, the possible values of X and Y are 0, 1, 2, 3, and 4.
2. To calculate the joint PMF, we need to determine the probability of each combination of X and Y occurring. For example, the joint probability P(X=0 and Y=0) represents the probability of getting zero l's and zero 2's. This can be calculated by multiplying the individual probabilities of getting zero l's and zero 2's in one roll, which is (5/6) multiplied by (5/6), giving us (25/36).
3. We repeat this process for all possible combinations of X and Y, calculating the joint probabilities for each.

By following these steps, we can create a table or matrix that shows the joint PMF values for each combination of X and Y.