Question

The probability mass function \( P(x) \) of a discrete random variable \( X \) is given by \( P(x) = \frac{1}{2^x} \), where \( x = 1, 2, \dots, \infty \). The expected value of \( X \) is _________.

• A.
• B.
• C.
• D.

Hint:

For a geometric distribution with \( P(x) = \frac{1}{2^x} \), the expected value is 2.

Answer 1

The Correct Option is B

The expected value \( E[X] \) of a discrete random variable is given by: \[ E[X] = \sum_{x=1}^{\infty} x P(x). \] Substituting the given probability mass function \( P(x) = \frac{1}{2^x} \), we get: \[ E[X] = \sum_{x=1}^{\infty} x \cdot \frac{1}{2^x}. \] This is a standard series that can be evaluated as: \[ E[X] = 2. \] Thus, the expected value of \( X \) is 2.