Question 19

Let $X$ be a Gaussian random variable with zero mean and variance $\sigma^2$, so $X \sim \mathcal{N}(0, \sigma^2)$. By symmetry of the normal distribution:

The random variable $Y$ is defined as $Y = \max(X, 0)$:

• When $X \le 0$, $Y = 0$.
• When $X > 0$, $Y = X > 0$.

Thus, the probability mass at $Y = 0$ is:

For any value $y < 0$, $P(Y \le y) = 0$. At $y = 0$, $P(Y \le 0) = P(Y = 0) = 0.5$.

By definition, the median $m$ of a random variable satisfies $P(Y \le m) \ge 0.5$ and $P(Y \ge m) \ge 0.5$. Since $P(Y \le 0) = 0.5$ and $P(Y \ge 0) = 1 \ge 0.5$, the median of $Y$ is precisely 0.