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: Expected Value Formula.
The expected value \( E(X) \) for a discrete random variable is calculated as the sum of each possible value multiplied by its probability: \[E(X) = \sum x_i \cdot P(x_i)\]Here, \( x_i \) represents the values the random variable can take, and \( P(x_i) \) is the probability associated with each value.

Step 2: Value Substitution.
Given the following values for \( X \) and their probabilities:- \( x_1 = -1 \), with \( P(x_1) = \frac{1}{8} \)- \( x_2 = 0 \), with \( P(x_2) = \frac{1}{2} \)- \( x_3 = 2 \), with \( P(x_3) = \frac{3}{8} \)Substitute these into the expected value formula:\[E(X) = (-1) \cdot \frac{1}{8} + 0 \cdot \frac{1}{2} + 2 \cdot \frac{3}{8}\]Perform the calculations:\[E(X) = \frac{-1}{8} + 0 + \frac{6}{8}\]\[E(X) = \frac{5}{8} = 0.625\]

Step 3: Final Result.
The calculated expected value of \( X \) is 0.625. This value corresponds to option (C).