GATE CS 2017 Set 1 Q19 — Probability: Random Variables and Distributions

We are given that $X$ is a Gaussian (normal) random variable with mean $\mu = 0$ and variance $\sigma^2$. Since the normal distribution is symmetric about its mean:

The random variable $Y$ is defined as $Y = \max(X, 0)$. Let us find the cumulative distribution function (CDF) of $Y$, denoted as $F_Y(y) = P(Y \le y)$:

• For $y < 0$, since $Y$ is always non-negative ($\max(X, 0) \ge 0$), we have $F_Y(y) = 0$.
• For $y \ge 0$, $Y \le y \iff \max(X, 0) \le y \iff X \le y$. Thus, $F_Y(y) = P(X \le y)$.

By definition, a median $m$ of a random variable $Y$ is any value that satisfies:

Let's evaluate these probabilities for $m = 0$:

1. First condition: $P(Y \le 0) = P(X \le 0) = 0.5 \ge 0.5$.
2. Second condition: Since $Y = \max(X, 0)$, $Y$ is never negative. Thus, $P(Y \ge 0) = 1 \ge 0.5$.

Since $m = 0$ satisfies both conditions, the median of $Y$ is 0.