Question

Suppose that the random variable \( X \) takes on the values: -1, 0, and 2 with probabilities \( \frac{1}{8} \), \( \frac{1}{2} \), and \( \frac{3}{8} \) respectively. Find the expected value of \( X \).

• A. 6
• B. 5.8
• C. 3.7
• D. 1

Hint:

The expected value is the sum of each possible value of the random variable weighted by its probability.

Answer 1

The Correct Option is C

Step 1: Formula for expected value.
The expected value \( E(X) \) of a discrete random variable is given by: \[ E(X) = \sum x_i \cdot P(x_i) \] where \( x_i \) are the values of the random variable, and \( P(x_i) \) are the corresponding probabilities.

Step 2: Substitute the given values.
We have the following values for \( X \): - \( x_1 = -1 \), \( P(x_1) = \frac{1}{8} \) - \( x_2 = 0 \), \( P(x_2) = \frac{1}{2} \) - \( x_3 = 2 \), \( P(x_3) = \frac{3}{8} \) Substituting these into the formula: \[ E(X) = (-1) \cdot \frac{1}{8} + 0 \cdot \frac{1}{2} + 2 \cdot \frac{3}{8} \] \[ E(X) = \frac{-1}{8} + 0 + \frac{6}{8} \] \[ E(X) = \frac{5}{8} = 0.625 \]

Step 3: Conclusion.
The expected value of \( X \) is 0.625, which is closest to option (C).