Question-76
The unit interval \((0,1)\) is divided at a point chosen uniformly distributed over \((0,1)\) in \(\mathbb{R}\) into two disjoint subintervals. Find the expected length of the subinterval that contains \(0.4\) rounded off to two decimal places.
\(0.74\)
Let the length of the sub-interval be \(\displaystyle L\). Then:
\[ L=\begin{cases} 1-X, & X< 0.4\\ X & X\geqslant 0.4 \end{cases} \]
Since the PDF of the uniform distribution \(\displaystyle U[ 0,1]\) in its support is just the constant \(\displaystyle 1\), we can compute the expectation of \(\displaystyle L\) as:
\[ \begin{aligned} \mathbb{E}[ L] & =\int\limits _{0}^{0.4} 1\cdot ( 1-x) dx+\int\limits _{0.4}^{1} 1\cdot xdx\\ & \\ & =\left[ x-\frac{x^{2}}{2}\right]_{0}^{0.4} +\left[\frac{x^{2}}{2}\right]_{0.4}^{1}\\ & \\ & =0.74 \end{aligned} \]