GATE CS 1999 Q1.1 — Probability: Random Variables and Distributions

The expectation $E[X]$ of a random variable $X$ represents its weighted average. By definition, a random variable cannot always take values strictly less than its expected value.

Suppose for contradiction that $X(\omega) < 5$ for all sample points $\omega$ in the sample space. Then, the expected value must also be strictly less than $5$, i.e., $E[X] < 5$. Since we are given $E[X] = 5$, this is a contradiction. Therefore, there must exist at least one sample point $\omega$ where $X(\omega) \ge 5$.

We can rule out the other options with simple counterexamples:

• Option A is false: If $X$ takes values $4$ and $6$ with equal probability $0.5$, then $E[X] = 5$, but $X$ never takes the value $5$.
• Option B is false: If $X$ takes the value $5$ with probability $1$, then $E[X] = 5$, but $X$ is never strictly greater than $5$.