Let $X$ be a Gaussian random variable with mean $0$ and variance $\sigma^2$. Let $Y = \max(X, 0)$ where $\max(a,b)$ is the maximum of $a$ and $b$. The median of $Y$ is ________.

Since $X \sim \mathcal{N}(0, \sigma^2)$ is symmetric about $0$:

Now consider the transformed random variable $Y = \max(X, 0)$:

• For $X \le 0$, $Y = 0$.

• For $X > 0$, $Y = X > 0$.

The cumulative distribution function (CDF) of $Y$, $F_Y(y) = P(Y \le y)$, satisfies:

• For $y < 0$: $F_Y(y) = 0$

• For $y = 0$: $F_Y(0) = P(Y \le 0) = P(X \le 0) = 0.5$

• For $y > 0$: $F_Y(y) = P(X \le y) > 0.5$

By definition, a median $m$ of $Y$ must satisfy $P(Y \le m) \ge 0.5$ and $P(Y \ge m) \ge 0.5$.

At $m = 0$:

Thus, the median of $Y$ is 0.