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+\frac12+q^2`

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-(x^2)/(2)|^(q)_0+|(x^2)/(2)|^(1)_(q)\\\\=(q-(q^2)/(2))+(1)/(2)-(q^2)/(2)=q+\frac12+q^2`

The expected length of the segment that contains Q
`=q+\frac12+q^2`