Let $X$ be a Gaussian random variable with mean 0 and variance $\sigma ^{2}$. Let $Y$ = max($X$,0)…

Question

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 ______________ .

Accepted Answer

Correct answer: 0 — Median = 0. There is a point mass at 0: P(Y = 0) = P(X ≤ 0) = 1/2 because X is symmetric with mean 0. If m < 0 then P(Y ≤ m) = 0 < 1/2, so no negative m can be a median. If m > 0 then P(Y ≤ m) = P(X ≤ m) = Φ(m/σ) > 1/2 but P(Y ≥ m) = P(X ≥ m) = 1 - Φ(m/σ) < 1/2, so no positive m can be a median. Therefore the unique median is 0, since it satisfies P(Y ≤ 0) = 1/2 and P(Y ≥ 0) = 1 ≥ 1/2.

Let \(X\) be a Gaussian random variable with mean 0 and variance \(\sigma…