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

Answer: (B), (C)

0 is the median of $X$

Answer 2

The correct option is (B): 0 is the median of $X$,(C): $M(t) = M(-t)$ for all $t \in (-1, 1)$