Question

For the probability distribution of \( X \) given below, the variance of \( X \) is

• A. 2.4257
• B. 2.5427
• C. 2.5742
• D. 2.2475

Hint:

For variance calculations, always use the formula \( \sigma^2 = E(X^2) - [E(X)]^2 \), and ensure correct substitution of values for expected value and squared values.

Answer 1

The Correct Option is D

Step 1: Use the formula for the variance.
The variance \( \sigma^2 \) of a probability distribution is given by: \[ \sigma^2 = E(X^2) - [E(X)]^2 \] where \( E(X) \) is the expected value of \( X \).
Step 2: Calculate \( E(X) \).
The expected value \( E(X) \) is given by: \[ E(X) = \sum_{i} P(X = x_i) \cdot x_i \] Substitute the values of \( X \) and \( P(X) \) into the formula to calculate \( E(X) \).
Step 3: Calculate \( E(X^2) \).
Similarly, \( E(X^2) \) is calculated as: \[ E(X^2) = \sum_{i} P(X = x_i) \cdot x_i^2 \] Substitute the values of \( X \) and \( P(X) \) into the formula to calculate \( E(X^2) \).
Step 4: Calculate the variance.
Substitute the calculated values of \( E(X) \) and \( E(X^2) \) into the variance formula to get the variance: \[ \sigma^2 = 2.2475 \]
Step 5: Conclusion.
Thus, the variance of \( X \) is 2.2475.