Question

A fair die is rolled twice independently. Let \( X \) and \( Y \) denote the outcomes of the first and second roll, respectively. Then \[ E(X + Y \mid (X - Y)^2 = 1) \] The value of \( E(X + Y \mid (X - Y)^2 = 1) \) is _________ (round off to 2 decimal places).

Hint:

When given conditional expectations, list all the possible outcomes that satisfy the condition and then calculate the mean of the corresponding values.

Answer 1

Correct Answer: 7

We are given that \( (X - Y)^2 = 1 \). This means that \( X - Y = \pm 1 \), so the possible pairs for \( (X, Y) \) are:
\[ (X, Y) = (2, 1), (3, 2), (4, 3), (5, 4), (6, 5) \quad \text{or} \quad (X, Y) = (1, 2), (2, 3), (3, 4), (4, 5), (5, 6). \] Thus, the possible values of \( X + Y \) are: \[ 3, 5, 7, 9, 11. \] The expected value \( E(X + Y \mid (X - Y)^2 = 1) \) is the average of these values: \[ E(X + Y \mid (X - Y)^2 = 1) = \frac{3 + 5 + 7 + 9 + 11}{5} = 7. \] Thus, the value is \( 7 \).