From the question on this page :
We break a unit length rod into two pieces at a uniformly chosen point. Find the expected length of the smaller piece
In one of the answers, the following reasoning is provided to begin with:
The length function is $f(x)=min(x,1−x)$ as you found. This is $x$ on the interval $[0,0.5]$ and $1−x$ on the interval $[0.5,1]$
Since the length of the stick is $x$, shouldn't we have $E(X) =\int_{x=0}^1 x f(x)dx = \int_{x=0}^1 x^2 dx = \frac{1}{3} $ ? Intuitively, that does not make sense but I would like to know what is amiss.
Why do we have the length function defined as $f(x)=min(x,1−x)$ and why is it $x$ on the interval $[0,0.5]$ and $1−x$ on the interval $[0.5,1]$ ?