Suppose you break a stick of unit length at a point chosen uniformly at random. Then the expected length of the shorter stick is ________ .

Let the break point $X$ be uniformly distributed over $[0, 1]$, so its probability density function is $f(x) = 1$ for $0 \le x \le 1$.

The length of the two pieces are $X$ and $1 - X$. The length of the shorter stick is:

The expected value of $L$ is:

Thus, the expected length of the shorter stick is 0.25.