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 _________.
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 _________.
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For a geometric distribution with \( P(x) = \frac{1}{2^x} \), the expected value is 2.
Updated On: Dec 26, 2025
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The Correct Option is B
Solution and Explanation
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.
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