GATE CS 2025 Set 2 Q.65: Probability and Statistics

The unit interval $(0, 1)$ is divided at a point chosen uniformly distributed over $(0, 1)$ in $\mathbb{R}$ into two disjoint subintervals.

The expected length of the subinterval that contains $0.4$ is ___________. (rounded off to two decimal places)

Your answer

Official answer and solution

Accepted answers: 0.7 to 0.8
From: Official GATE 2025 answer key

Solution

Express the length $L(X)$ of the subinterval containing $0.4$ as a piecewise function of $X \sim \text{Uniform}(0, 1)$ and integrate over $(0, 1)$ .

Let $X \sim \text{Uniform}(0, 1)$ be the random cut point, which divides $(0, 1)$ into $(0, X)$ of length $X$ and $(X, 1)$ of length $1 - X$ . Let $L$ be the length of the subinterval containing $0.4$ :

If $0 < X < 0.4$ , then $0.4 \in (X, 1)$ , so $L = 1 - X$ .
If $0.4 < X < 1$ , then $0.4 \in (0, X)$ , so $L = X$ .

Since $X$ has uniform density $f_X(x) = 1$ on $(0, 1)$ :

E[L] = \int_0^{0.4} (1 - x)\,dx + \int_{0.4}^1 x\,dx = \left[x - \frac{x^2}{2}\right]_0^{0.4} + \left[\frac{x^2}{2}\right]_{0.4}^1

E[L] = (0.4 - 0.08) + (0.50 - 0.08) = 0.32 + 0.42 = 0.74

Final answer

0.74

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

Source

Paper: GATE 2025 Computer Science and Information Technology - Session 2 (CS2)
Question: 65 of 65
Section: Computer Science and Information Technology

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