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}$
The above expression is a martingale, just use Ito calulus to produce a formula that does not include an integral which is integrating w.r.t time, hence it does not change with a change in time. Therefore the expected value of any martingale is 0.