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?

• A. \( P(X = -X) = 1 \)
• B. 0 is the median of \( X \)
• C. \( M(t) = M(-t) \) for all \( t \in (-1, 1) \)
• D. \( E(X) = 1 \)

Answer 1

The Correct Option is B, C

1. Symmetry of \( f(x) \): - Since \( f(x) = f(-x) \), the random variable \( X \) is symmetric about 0. 2. Median of \( X \): - For symmetric distributions, the median coincides with the point of symmetry. Thus, the median of \( X \) is 0. 3. Moment Generating Function: - Symmetry implies \( M(t) = E(e^{tX}) = E(e^{-tX}) = M(-t) \). 4. Expectation of \( X \): - Due to symmetry, \( E(X) = 0 \), not 1. 5. Probability P(X = -X): - For continuous random variables, \( P(X = -X) = 0 \), as it is the probability of a single point.