Let X be a continuous random variable whose cumulative distribution function (CDF) F_X(x), for some t, is given as follows: F_X(x) = cases 0 & x ≤ t \\ x - t4 - t & t ≤ x ≤ 4 \\ 1 & x ≥ 4 cases If the median of X is 3, then what is the value of t?

GATE DA 2025 Q19: Uniform CDF on [t, 4] with median 3: find t

Let $X$ be a continuous random variable whose cumulative distribution function (CDF) $F_X(x)$ , for some $t$ , is given as follows:

$F_X(x) = \begin{cases} 0 & x \le t \\ \dfrac{x - t}{4 - t} & t \le x \le 4 \\ 1 & x \ge 4 \end{cases}$

If the median of $X$ is 3, then what is the value of $t$ ?

Try it: multiple choice: exactly one correct option

Wrong answer costs 1/3 mark in GATE.

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Answer: A (official GATE 2025 final answer key)

Concept used: The median $m$ satisfies $F_X(m) = 0.5$ .

First step: Set $F_X(3) = \dfrac{3 - t}{4 - t} = \dfrac{1}{2}$ and solve for $t$ .

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Source: official GATE 2025 DA question paper and final answer key, organised by IIT Roorkee. Question paper · Answer key