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How can we easily compute $\mathbb{E} [ \left|W_t\right|]$, where $W = (W_t)_{t \geq 0}$ is the one dimensional standard Brownian motion (or wiener process)?

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up vote 5 down vote accepted

The finite one-dimensional distribution of $W_t$ is normal with mean 0 and variance $t$. This means that $E[|W_t|]=\sqrt{\frac{2t}{\pi}}$ because $$\int_0^{\infty} 2 x \frac{1}{\sqrt{2 \pi t}} \exp \left( - \frac{1}{2 t} x^2 \right) \mathrm{d} x = \sqrt{\frac{2t}{\pi}}$$

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