Question

If a random variable \( X \) has the following probability distribution, then its variance is nearly: 

• A. \( 2.8875 \)
• B. \( 2.9875 \)
• C. \( 2.7865 \)
• D. \( 2.785 \)

Hint:

To find the variance of a probability distribution, compute \( E(X^2) \) and use \( \text{Var}(X) = E(X^2) - (E(X))^2 \).

Answer 1

The Correct Option is A

Step 1: Verify that probabilities sum to 1
We have: \[ 0.05 + 0.1 + 2K + 0 + 0.3 + K + 0.1 = 1. \] Solving for \( K \): \[ 2K + K = 1 - (0.05 + 0.1 + 0.3 + 0.1) = 1 - 0.55 = 0.45. \] \[ 3K = 0.45 \Rightarrow K = 0.15. \] Step 2: Compute Expectation \( E(X) \)
\[ E(X) = \sum x P(X=x). \] \[ = (-3)(0.05) + (-2)(0.1) + (-1)(2K) + (0)(0) + (1)(0.3) + (2)(K) + (3)(0.1). \] \[ = (-0.15) + (-0.2) + (-0.3) + 0 + 0.3 + 0.3 + 0.3 = 0. \] Step 3: Compute \( E(X^2) \)
\[ E(X^2) = \sum x^2 P(X=x). \] \[ = (-3)^2(0.05) + (-2)^2(0.1) + (-1)^2(2K) + (0)^2(0) + (1)^2(0.3) + (2)^2(K) + (3)^2(0.1). \] \[ = (9)(0.05) + (4)(0.1) + (1)(0.3) + 0 + (1)(0.3) + (4)(0.15) + (9)(0.1). \] \[ = 0.45 + 0.4 + 0.3 + 0.3 + 0.6 + 0.9 = 2.8875. \] Step 4: Compute Variance \( \sigma^2(X) \)
\[ \text{Var}(X) = E(X^2) - (E(X))^2. \] Since \( E(X) = 0 \), \[ \text{Var}(X) = 2.8875 - 0^2 = 2.8875. \] Step 5: Conclusion
Thus, the final answer is: \[ \boxed{2.8875}. \]