Laplace transform of the integrated square of Brownian motion
= Laplace transform of the integrated square of Brownian motion
{c}
{title2=$\mathbb E e^{-\lambda Q}=(\cosh\sqrt{2\lambda})^{-1/2}$}
The independent squared-normal series for the <integrated square of Brownian motion> gives the product $\prod_n(1+2\lambda/((n-1/2)^2\pi^2))^{-1/2}$. The half-integer cosine product reduces this to $(\cosh\sqrt{2\lambda})^{-1/2}$. Expanding at zero gives mean $1/2$ and variance $1/3$.