To find the expected value of a random variable, you multiply each score by its corresponding probability and then sum up the products. The formula is:

$E(X) = \sum (x_i \cdot P(x_i))$

Where $x_i$ are the scores and $P(x_i)$ are their corresponding probabilities.

From the table:

• For $x_1 = 0, P(x_1) = 0.12$
• For $x_2 = 1, P(x_2) = 0.23$
• For $x_3 = 5, P(x_3) = 0.42$
• For $x_4 = 8, P(x_4) = 0.09$
• For $x_5 = 14, P(x_5) = 0.14$

Now, calculate:

$E(X) = (0 \cdot 0.12) + (1 \cdot 0.23) + (5 \cdot 0.42) + (8 \cdot 0.09) + (14 \cdot 0.14)$

$E(X) = 0 + 0.23 + 2.1 + 0.72 + 1.96$

$E(X) = 4.99$

The expected value of the random variable is 4.99.

Do you have any questions or need further clarification?

Related Questions:

1. What is the significance of expected value in probability theory?
2. How would the expected value change if the probabilities were altered?
3. Can the expected value ever be negative?
4. What are real-world examples where expected value calculations are used?
5. How is the variance calculated for a discrete random variable?

Tip:

Always verify that the sum of probabilities equals 1 to ensure the calculations are based on a valid probability distribution.