Final Answer:
a. The sum of the two numbers
b. The product of the two numbers
c. E(x) * E(y) because, when dealing with independent random variables, the expected value of their product is equal to the product of their individual expected values.
Step-by-step explanation:
The random variable x represents the sum of the two numbers when a die is rolled twice, making option b (The sum of the two numbers) the correct choice for x. On the other hand, the random variable y represents the product of the two numbers, making option a (The product of the two numbers) the correct choice for y.
The expected value of the product xy is equal to the product of the expected values E(x) and E(y), leading to the correct choice c (E(x) * E(y)). This is a property of the expected value operator when dealing with independent random variables. Mathematically, E(xy) = E(x) * E(y) holds true when x and y are independent, meaning the outcome of one roll does not affect the outcome of the other. This relationship allows us to calculate the expected value of the product of two random variables based on the individual expected values, providing a useful tool in probability theory and statistics.
In summary, when rolling a die twice, x represents the sum of the two numbers, y represents the product of the two numbers, and the expected value of their product (E(xy)) is equal to the product of their individual expected values (E(x) * E(y)) under the assumption of independence.