The unit interval (0,1) is divided at a point U chosen uniformly distributed over (0,1) into two disjoint subintervals. The expected length of the subinterval that contains 0.6 is __________ (rounded off to two decimal places).

GATE 2025 · Engineering Mathematics · Uniform Distribution · medium

Answer: 0.74

1. Define the piecewise length function: When U = u < 0.6, the subinterval (u, 1) contains 0.6 with length 1 - u. When U = u > 0.6, the subinterval (0, u) contains 0.6 with length u.
2. Set up the expectation integral: Since U ~ Uniform(0,1) has density f(u) = 1, E[L] = integral_0^0.6 (1-u) du + integral_0.6^1 u du.
3. Evaluate the first integral: integral_0^0.6 (1-u) du = [u - u^2/2] from 0 to 0.6 = (0.6 - 0.36/2) - 0 = 0.6 - 0.18 = 0.42.
4. Evaluate the second integral: integral_0.6^1 u du = [u^2/2] from 0.6 to 1 = 1/2 - (0.6)^2/2 = 0.5 - 0.18 = 0.32.
5. Add both parts: E[L] = 0.42 + 0.32 = 0.74.