Let $\mu$ be a probability distribution on the interval $[0,1]$ with probability density function $p(x)=c \cdot x^{2}$ where $c$ is an undetermined constant. Consider an interval $A=[a, b]$, with $0 \leq a<b\leq1$ such that
$$\text{Pr}_{X \sim \mu}^{}[X \in A]=\int_{a}^{b} p(x) d x=\frac{1}{2} .$$
Then, the smallest possible value of $b-a$ is
- $1-\frac{1}{2^{1 / 3}}$
- $\frac{1}{2^{1 / 3}}$
- $1-\frac{1}{2^{1 / 2}}$
- $\frac{1}{2^{1 / 2}}$
- The smallest possible value of $b-a$ cannot be determined uniquely from the information given in the question.
