Question

Two fair dice are rolled, and the random variable \( X \) denotes the sum of the outcomes. What is the expected value of \( X \)?

• A. 6.5
• B. 7
• C. 7.5
• D. 8

Hint:

For any two independent random variables \( X \) and \( Y \), the expectation follows the additive property: \[ E[X + Y] = E[X] + E[Y] \] This property simplifies expectation calculations in probability theory.

Answer 1

The Correct Option is B

Expectation of the Sum of Two Dice.
Let \( X_1 \) and \( X_2 \) be the outcomes of two fair six-sided dice. The sum of outcomes is: \[ X = X_1 + X_2 \]
For a single fair die: \[ E[X_1] = E[X_2] = \frac{1+2+3+4+5+6}{6} = \frac{21}{6} = 3.5 \]
By linearity of expectation: \[ E[X] = E[X_1] + E[X_2] = 3.5 + 3.5 = 7 \]
Thus, the expected value of \( X \) is: \[ \mathbf{7} \]