Question

Let \( X \) be the discrete random variable representing the number (\( x \)) appeared on the face of a biased die when it is rolled. The probability distribution of \( X \) is as follows: \[ X = x: \quad 1, \, 2, \, 3, \, 4, \, 5, \, 6 \] \[ P(X = x): \quad 0.1, \, 0.15, \, 0.3, \, 0.25, \, k, \, k \] The variance of \( X \) is:

• A. \( 1.64 \)
• B. \( 1.93 \)
• C. \( 2.16 \)
• D. \( 2.28 \)

Hint:

To calculate the variance, ensure you compute both \( \mathbb{E}(X^2) \) and \( (\mathbb{E}(X))^2 \). Double-check the consistency of the probability distribution.