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

My try:
Somewhere it explain as: Here, half of the values of Y are to the left of the mean X = 0 and the remaining half of the values of Y lies to the right of the mean X = 0. hence,The median of Y = 0.
Can you please explain in other way?
You have for $X$ a continuous random variable with median $0$


$P(X \le 0)=\frac12$
$P(X \gt 0)=\frac12$


so, since $X \le 0 \implies Y=0$ and $X\gt 0 \implies Y \gt 0$,


$P(Y =0)=\frac12$
$P(Y \gt 0)=\frac12$


so, moving towards the definition of a median, 


$P(Y \le 0) \ge \frac12$
$P(Y \ge 0) \ge \frac12$


so the median of $Y$ is $0$
As mean =0 and variance  σ2 represents the standard normal distribution.And the graph of Standard normal distribution is symmetric And in symmetric we have,
Mean=Median=Mode.

As given that mean is 0.

Hence Median and mode will be 0