Question

Let X and Y each have the distribution of a fair six-sided die rolled once, and let Z= X +Y. = What is E(X | Z]? Express your answer in terms of Z (no need to use LaTeX).

Answer 1

To find E(X | Z), we need to find the conditional expected value of X given that Z = z. The conditional expected value of X is equal to z / 2.

Step-by-step explanation:

To find E(X | Z), we need to find the conditional expected value of X given that Z = z. In this case, Z represents the sum of two fair six-sided dice rolled once, and X represents the value of the first die. Since the dice are fair, each side has a probability of 1/6 of being rolled.

Let's calculate E(X | Z = z):

1. When z = 2, there is only one possible outcome: X = 1. Therefore, E(X | Z = 2) = 1.
2. When z = 3, there are two possible outcomes: X = 1 or X = 2. The expected value in this case is (1 + 2) / 2 = 1.5.
3. When z = 4, there are three possible outcomes: X = 1, X = 2, or X = 3. The expected value is (1 + 2 + 3) / 3 = 2.
4. Continuing this calculation, we can find the expected value for other values of z.

Therefore, the conditional expected value of X given Z = z is z / 2. In other words, E(X | Z) = Z / 2.

Answer 2

The conditional expectation, E(X | Z), can be calculated by finding the conditional probability distribution of X given Z. This can be done by considering all possible combinations of X and Y that sum to each value of Z. Once the conditional probability distribution is obtained, the conditional expectation can be calculated by taking the sum of X multiplied by its corresponding conditional probability for each value of Z.

Step-by-step explanation:

The question asks for the conditional expectation, E(X | Z), where X and Y are random variables representing the outcomes of rolling a fair six-sided die once and Z is the sum of X and Y. In order to calculate E(X | Z), we need to understand the distribution of X given the value of Z. Since X and Y each have the distribution of a fair six-sided die, they can take on values from 1 to 6 with equal probability. When we sum two dice rolls, the possible values of Z range from 2 to 12. We need to find the conditional probability distribution of X given Z.

For each possible value of Z, we need to find the probability distribution of X conditioned on that value. For example, if Z is 2, that means X + Y = 2. The only possible values for X and Y in this case are (1,1) since the only combination that sums to 2 is 1 + 1. Therefore, P(X = 1 | Z = 2) = 1. Similarly, for Z = 3, there are two possible combinations (1,2) and (2,1), so P(X = 1 | Z = 3) = 1/2 and P(X = 2 | Z = 3) = 1/2. By calculating these conditional probabilities for each value of Z, we can construct the conditional probability distribution of X given Z. Once we have the conditional probability distribution, we can calculate E(X | Z) by taking the sum of X multiplied by its corresponding conditional probability for each value of Z.