Let $Y = \max(X, 0)$, then median of $Y$ is _____.

Question

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?

Answer 1

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 $0$

Answer 2

The median of $Y$ is defined as the least value $m$ such that $\mathsf P(Y\leqslant m)\geqslant\tfrac 12$.

Since $Y=\max\{0,X\}$, then: $\mathsf P(Y\leqslant 0){~=~\mathsf P(Y=0)\\~=~ \mathsf P(X\leqslant 0)\\~=~\tfrac 12}$

... since $X$ is symmetrically distributed about $0$.

Thus $0$ is the median of $Y$.

That is all there is too it.