Question:
Let \( X \) be a continuous random variable with a probability density function \( f \) and the moment generating function \( M(t) \). Suppose that \( f(x) = f(-x) \) for all \( x \in \mathbb{R} \) and the moment generating function \( M(t) \) exists for \( t \in (-1, 1) \). Then which of the following statements is/are correct?
Let \( X \) be a continuous random variable with a probability density function \( f \) and the moment generating function \( M(t) \). Suppose that \( f(x) = f(-x) \) for all \( x \in \mathbb{R} \) and the moment generating function \( M(t) \) exists for \( t \in (-1, 1) \). Then which of the following statements is/are correct?
Updated On: Oct 1, 2024
- \( P(X = -X) = 1 \)
- 0 is the median of \( X \)
- \( M(t) = M(-t) \) for all \( t \in (-1, 1) \)
- \( E(X) = 1 \)
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The Correct Option is B, C
Solution and Explanation
The correct option is (B): 0 is the median of \( X \),(C): \( M(t) = M(-t) \) for all \( t \in (-1, 1) \)
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