GATE CS 2025 Set 1 Q.58: Probability and Statistics

Consider a probability distribution given by the density function $P(x)$ .

P(x) = \begin{cases} Cx^2, & \text{for } 1 \le x \le 4 \\ 0, & \text{for } x < 1 \text{ or } x > 4 \end{cases}

The probability that $x$ lies between 2 and 3, i.e., $P(2 \le x \le 3)$ is __________. (rounded off to three decimal places)

Your answer

Official answer and solution

Solution

Find the normalization constant $C$ from $\int_1^4 Cx^2\,dx = 1$ and integrate $P(x)$ from $2$ to $3$ .

\int_{1}^{4} C x^2 \, dx = C \left[\frac{x^3}{3}\right]_{1}^{4} = C \cdot \frac{64 - 1}{3} = 21C = 1 \implies C = \frac{1}{21}

P(2 \le x \le 3) = \int_{2}^{3} \frac{1}{21} x^2 \, dx = \frac{3^3 - 2^3}{63} = \frac{27 - 8}{63} = \frac{19}{63} \approx 0.301587 \approx 0.302

Final answer

0.302

The answer is the one in the official GATE answer key. The solution is this site's own explanation.

Paper: GATE 2025 Computer Science and Information Technology - Session 1 (CS1)
Question: 58 of 65
Section: Computer Science and Information Technology

GATE questions are the work of the organising institutes. This site is independent and is not affiliated with them.