# expected value of brownian motion

How can you find this expected value? $$\mathbb{E}[|W_{t}^2 - t|]$$

where $W_{t}$ is a brownian motion.

$W_t$ is a normal random variable with mean $0$ and variance $t$. If $f(x)$ is the density of a standard normal distribution, you're looking at $t \int_{-\infty}^\infty |x^2 - 1| f(x)\ dx$, which according to Maple is $2 t e^{-1/2} \sqrt{2/\pi}$