TIFR ECE 2017 | Question: 11

Question

Consider a unit length interval $[0,1]$ and choose a point $X$ on it with uniform distribution. Let $L_{1}=X$ and $L_{2}=1-X$ be the length of the two sub-intervals created by this point on the unit interval. Let $L=\max \left\{L_{1}, L_{2}\right\}$. Consider the following statements where $\mathbf{E}$ denotes expectation.

1. $\mathbf{E}[L]=3 / 4$
2. $\mathbf{E}[L]=2 / 3$
3. $L$ is uniformly distributed over $[1 / 2,1]$
4. $L$ is uniformly distributed over $[1 / 3,1]$

Which of the above statements is/are $\text{TRUE?}$ Choose from the following options.

• A. Only $\text{(i)}$
• B. Only $\text{(ii)}$
• C. Only $\text{(i)}$ and $\text{(iii)}$
• D. Only $\text{(ii)}$ and $\text{(iv)}$
• E. None of the above