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

Correct Answer: 0.7

Consider a point chosen at random on the interval \((0,1)\), which divides the interval into two parts. The point \(0.4\) will always lie in one of these two parts.

If the dividing point lies to the right of \(0.4\), then the segment containing \(0.4\) stretches from \(0\) to the dividing point. If it lies to the left of \(0.4\), then the segment containing \(0.4\) extends from the dividing point to \(1\).

In either situation, the length of the segment containing \(0.4\) is simply the larger of the two pieces formed. Averaging this length over all possible positions of the dividing point leads to a value lying between \(0.70\) and \(0.80\).