A stick is broken into 3 pieces, by randomly choosing two points along its unit length, and cutting it. What is the expected length of the middle part? Hint
Selecting the random point from a small 'dt' length element is dt , as length of stick=1. Now use the definition of Expectation.
Double integral of |x-y|dxdy gives 1/3 as answer. This is same as one would expect from a broken pencil.
Integrate from 0 to 1, x*x/2 + (1-x)*(1-x)/2 = 1/3
logic: if one cut is at distance x from left, with probability x, the second cut comes before it, and expected length of middle piece is x/2.. Similarly with prob (1-x) it, middle piece is expected to have length (1-x)/2. Thus adding and integrating from 0 to 1.