Question

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)

Hint:

For a uniform distribution, calculate expected lengths by considering the behavior of the interval boundaries.

Answer 1

Let the dividing point \(x\) be uniformly distributed over \((0, 1)\). The two subintervals formed are \((0, x)\) and \((x, 1)\). The subinterval that contains 0.4 will either be \((0, x)\) if \(x%gt;0.4\) or \((x, 1)\) if \(x%lt;0.4\). The length of the subinterval containing 0.4 is \(\max(x, 1 - x)\). Since \(x\) is uniformly distributed, the expected value of this maximum length is between 0.70 and 0.80.