Question

A point Q lies in the interval [0,1]. Suppose the distance between 0 and Q is q. Then a point is randomly selected from the interval [0,1], diving the line into two segments. What is the expected length of the segment that contains Q

Answer 1

Answer: $q+\dfrac12+q^2$

Step-by-step explanation:

Let y gives the length of the segment having Q.

Then, the other pat = 1-y

Now, the expected length of the segment that contains Q = $\int^{q}_0(1-x)dx+\int^{1}_{q}x dx\\\\=|x-\dfrac{x^2}{2}|^{q}_0+|\dfrac{x^2}{2}|^{1}_{q}\\\\=(q-\dfrac{q^2}{2})+\dfrac{1}{2}-\dfrac{q^2}{2}=q+\dfrac12+q^2$

The expected length of the segment that contains Q $=q+\dfrac12+q^2$