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GATE 2025 DA – Question 19

Probability and Statistics · Random variables, discrete and continuous distributions · 1 mark · Multiple choice

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

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

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

  1. 2
  2. 1
  3. -1
  4. 0

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Correct answer: (A) 2

Explanation

The median is the point where the CDF is $0.5$: $\frac{3 - t}{4 - t} = \frac{1}{2}$. Then $2(3 - t) = 4 - t$, so $6 - 2t = 4 - t$ and $t = 2$. (The variable is uniform on $[2, 4]$, whose midpoint is 3.)