An Explicit Formula for the C.D.F. of the $L_1$ Norm of the Brownian Bridge

Abstract

Let $X(t), 0 \leq t \leq 1$, be the Brownian bridge and $L = \int^1_0 |X(t)| dt$. Using results of Shepp [5] and Rice [4], an explicit formula for $P(L \leq \ell)$ is obtained.